I cannot comment on the possible scientific approaches, but their downfalls would be all the same: individual invisibility is great for spying, criminal activity and maybe small special operations teams.
Any scale of warfare needs a very detailed picture of where the friendlies are. You need to coordinate the troops. You need to evacuate the wounded. You need to make sure there is no friendly in the explosion radius or line of fire of your weapon. You do not want to miss on the morale boost that the feeling of "the man next to me" gives to the troops. In short, what you really don't want is an individual soldier, seemingly all alone, in danger of getting hit by the enemy as well as getting hit by any weapon his buddies may fire, fully aware that he cannot be evacuated and any wound will be his death. So what you need is a way to identify those invisible soldiers. And if that technology exist, it's only a matter of who wins the encryption race. Will the enemy be able to decrypt (or maybe just trace the signal to a specific point in 3D to fire on, that would be enough) the signal in real time and kill the invaders or not. Kind of the Enigma in WWII.
Generally speaking, technology can come in two variants:
- Fooling the individual viewer
- Actually sending a different image
Fooling the viewer by for example somehow manipulating his brain can easily cicurmvented by technology available today. Cameras. If you cannot fool an optical recording device, things like night vision or modern day tank optics will already invalidate the whole concept, because it's short time recording and replay.
So what you need is a device that actually changes the rays of light and transmit a different image to any image processing device, technical or biological. The obvious countermeasure is to emply a different sense for detection. Guard dogs are not exactly the pinnacle of technology, but they work. Ok, lets assume with the invisibility kit came a "smell transmutor". We could still detect moving bodies by air pressure. And radar. And maybe Infrared or Ultraviolet. Or maybe ultrasonic sound. So this invisibility kit needs to be a full blown scientific wonder, or the technology to detect invisible people will be cheaper than the invisibility itself.
Assuming the invisibility is perfect in any way, it would still be vulnerable to things that normally obscure visibility or hinder our senses. Invisible people moving through smoke or rain will be visible by their surroundings. The rain not hitting spots, the smoke moving around columns of "nothingness". Any kind of muddy ground can make technology worth a few millions totally meaningless. Comic-style even in the case of muddy footprints. People will need to be very skilled and specially trained even if they have perfect technology.
And still, if the guy employing such a kit is absolutely not detectable, the world is in danger of failing. How do you prevent crimes with such technology available. For a believable world, you will need a way to control such technology. If this technology is widely available in wartime, what would the civilization look like in peacetime? With all the guys that know that nothing can stop them? Maybe the real drawback of the technology will be, that after war all the veterans able to use that technology will need to vanish without trace because society would fail if they were actually coming back alive.