Let's be honest, swords are cool, we can all agree on this, but the invention of the bullet (which removed the musket's reloading time) has practically removed swords from any form of combat where guns are an option.
What can I do to make sure that soldiers in my culture still use swords with their guns? What changes must I make to history? Is it even possible to achieve this?
Unlike this question I am specifically asking for swords in modern scenarios,as opposed to general melee in the future.
I'm going preface this by saying that anything I don't back up with a source, I've learned by my experience and study as a Historic European Martial Artist (a HEMA practitioner, for the acronym minded). This answer has some background before actually proposing solutions, but that background helps to understand what you need to change.
There are actually modern situations where a sword would do much better than a gun. The police have a "21 foot rule." Basically, a person with a knife (or sword) wins against a person trying to draw a gun. Even with a gun drawn, 10 feet beats a gun.
Body armor for bullets does not mean body armor against knives or blades, as seen here. In fact, some companies which attempt to provide prisons guards armor need to change their armor to adapt to stabs. (As you can see here, this armor is specifically meant for protection against stabs, not bullets! They also sell different armor for ballistic protection because they're not the same thing!) Additionally, it's been known for a long time that you aim for the weak spots, so don't stab/shoot someone in the gut when you can go for the unprotected face! That makes this whole armor argument a much more tricky!
So swords in warfare? This isn't something we've seen for a long time, mostly because it's hard to get a large force within 21 feet of solders. As seen in this reddit post, where veterans and solders talk about it, bayonets and some knives still see use in modern warfare. These blades are usually used in urban areas, where you may not have time to aim your gun.
Can a sword do a little better than a bayonet or knife? Well yes; you will (generally) have increased reach, so you can kill the other guy before he reaches you. Additionally, swords can have more cutting (or thrusting) power than a knife, and can also be used in grappling.
Actually, this is a general trend in combat in general; if a weapon allows you to kill them before they can even reach you, or allows you to fight at the same distance, you use that weapon first! (Tactical considerations aside...) I'm calling this the "kill them at distance" (KTAD) principle.
Coming from a Historic European Martial Arts background, I know that swords were more commonly sidearms than primary weapons. The weapon of war before the rifle was, generally, a staff weapon. (Staff weapons include spears, lances, and other pole-arms.) This is because of the KTAD principle.
Pike formations were even common into the late renaissance, where muskets were common. The bayonet became popular because you can get some advantages of staff weapons but also retain your ability for long-distance combat. (Although some people say bayonets may not actually have seen much killing...) During the colonial age, many solders used swords as a backup weapon because they were often outnumbered and their gun was not up to the task of killing that many people before they were overrun.
In any case, as guns became easier to use and their rate of fire and capacity for armor penetration increased, people took the KTAD principle and selected to use the weapon that kills the enemy without putting themselves in mortal danger. Early in the days of muskets, people did still wear plate, and it did help against musket fire. Eventually, though, the power of the bullets were simply too much for the armor of the time, so the increased weight of the armor was a liability and therefore ditched.
You need lightweight, full-body armor that's resistant to high-kinetic-energy projectiles, aka bullets, but vulnerable to slicing (the shields in Dune, but without the handwaving). Although I suspect the first thing militaries would do to counter the armor is invest in a lot more artillery, not spend hundreds of hours training grunts to use swords...
What if duels were legal, but only with swords? If you kill the opponent on a proper duel you face no reprecussions.
Of course, trained person would easily defeat untrained person. If nobility was consistently trained then they would like how things are.The law is just, it allows self-defence for poor and rich alike. When guns are introduced, duels with guns would be outlawed (because power balance shifts in a way elites don't like). Guns could be also outlawed for self-defence too.
Then people will carry swords instead of guns: sword is a pretty convenient melee weapon to carry.
Of course, on the battlefield soldiers will carry guns if those are effective. What can be done about that?
Well, if everything happens in space then soldiers don't need to carry guns - stuff like boarding is heavily unlikely and they won't ever get to a personal gun's shooting range.
Navy people could easily carry swords though. Carrying a gun is pretty stupid in space - there's only your crewmates to shoot and if you miss both your target, you and people in immediate vicinity are going to have a bad time.
Heavy armor or shields, that are reactive to fast moving objects or objects with high energy can be countered by close range combat weapons like swords.
This method is probably most commonly seen in games like Mass Effect but there are also mentions in Frank Herbert's Dune in the form of shield belts.
Shields:
A simple way to realize such shields on todays technological basis would be some kind of electromagnetic shield (EM-shield). There is the concept of Eddy current brakes which use magnetic induction within conductive materials to transform kinect into heat energy. This concept is based on inhomogeneous magnetic fields. A fast moving object has to travel through this field and as an effect of the changing magnetic flow (due to the inhomogeneity) will then generate currents according to Lenz's law that counteract the change in magnetic flow. Those currents will (due to the electrical resistance) generate heat and reduce on basis of the conservation of energy the kinect energy of the moving object. A neat coincidence is that faster moving objects will have higher rates of kinect energy loss and thus experience the biggest "repulsion". This short video shows the concept, however here, the magnet is moving which in general can be interchanged by a stationary magnet and a moving conductive material.
With high enough field-strength, in principle every material can become conductive. There is also the possibilibty of incorporating fast changing electric fields. A static field can split up charges within an object and thus create measurable postive or negative charges on opposing sites. These in return can be used to repulse the object, however a change in the direction of the electric field has to be employed to do so. The object will react to the changed electric field polarity within a small time interval in which the electric field has to be changed again. This, however, needs active recognition of the object's material e.g. on basis of the induced fields to modulate the shield accordingly.
A general drawback of a repulsive shield system is that in general all systems have to obey to the conservation of momentum. That is, the shield systems will always generate a momentum, depending on the relative mass between the moving object and the shield system (e.g. light objects can be stoped more easily as heavy ones). This however, is another argument for fighting with swords instead of bullets.
Armor:
There are materials that react to fast moving objects differently than to slow ones. Take for instance non-newtonian fluids, that become much "harder" on impact of fast objects. Such materials could be used as armor layers to protect against fast moving objects while letting slow ones still pass. This has the advantage that slow movments (like body movements) will not trigger the protective effect and thus will not harm the wearer's mobility. However, the speed of a striking sword is in the range of usual body movements, making this type of armor less effective against swords.
There is also a material called Starlite (however, its inventor didn't publish its composition) which is told to be resistant against laser beams due to its great insulating capability: click for video
Swords fell out of use largely because they became more and more useless in war and culture was became more and more urban thus less and less people had use of it in their day to day lives. With the advent of guns this just sped up their demise as the weapon of choice in areas of life...
So how do you bring swords back in modern days?
So there is plenty of ways to possibly move back to swords, but you also then have to make sure it's not viable to use steel armor either cuz then swords are worthless again.
Swords were used until the end of the 19th century, one thing changed the deal in warfare, the Haber process. It's a chemical process that synthesize ammoniac the rarest component of explosif at the time.
With this process we extract ammoniac from the air, before it was a natural formation collect in moist caves. It maid explosive incredibly cheap and the production of ammunition exploded. This lead to the artillery based strategy of the WW1.
Without this process you cannot maintain a long conflict based on gun power, it will still be there and most of the kill will be by guns but you will have some cool actions with blades.
An another point on why they stop using swords is it need more training than guns :
The Haber Process was also a true revolution in agriculture since ammoniac is a used in fertilizer. Without it humanity would not have been able to grow has much and the demography would be very less high. This give you less people to do war. The point is with less soldiers you can't just throw untrained young people into the hell of war. You are more incline to train well your troops => give them a sword trainning.
Guns shoot bullets according to a maximum rate of fire. The faster that rate of fire becomes, the shorter the distance becomes between bullets. The fastest rate of fire there can be is one in which there is no space between bullets—i.e., the bullets are touching each other end-to-end as they exit the barrel. So now the weapon looks feels different. Instead of firing a series of projectiles, it is essentially a solid cylinder of bullets pressed together end-to-end. This is like a sword with a long, spaghetti noodle for a blade that is actually bullets moving incredibly fast and tearing through whatever the operator waves the gun at. You don't need to aim by pointing and shooting, but rather just a slice that causes the bullet string to sweep across the target, like a blade would.
Update in response to comments:
A machine like the sword-gun would consume an enormous amount of ammunition that its operation is somewhat prohibitive, but not impossible. It is not the kind of weapon that one soldier can carry into battle as a personal weapon, but rather it's like a stationary piece of artillery that is connected directly to the outlet of a supply column that serves it. Several guns stationed at artillery bases would allow operators of the device to slice through the gates of the city while laying siege from miles away. As many as hundreds of miles behind the front lines there are mobilized bullet factories operating at peak capacity to pump a continuous stream fresh bullets into bullet pipelines. These large pipes run over the surface of conquered lands directly to the front lines and into the sword-guns.
Use of the sword-gun has environmental concerns as well. After a city has been ravaged by these weapons there is an enormous amount of bullet metal buried deep within the scars of the city. It could take many months to mine out all of that material when the city is reconstructed after the battle.
