Ironically enough, these kinds of vehicles are actually less likely today than they were in (say) the 1960s. The reason for that is the current focus on efficiency.
When you get right down to it, anything that hovers has to constantly apply energy to maintaining a set distance from the ground. This is on top of any other energy that has to be applied to get the hovering vehicle to actually move. This is because of gravity; the Earth attracts all nearby smaller objects to itself and as such you need to counteract that force by turning real energy into potential energy (the energy that would be applied by the earth to a falling object) on a constant basis.
The wheel on the other hand actually uses gravity to its advantage by using weight to press the wheel to the ground, causing tension which in turn causes friction, which (if stronger than the rotational energy applied to the wheel via the axle) causes lateral movement through rolling. This is why car tyres are made of rubber; it maximises the friction between the tyre and the road, making the lateral movement of the car the least resistant application of the energy.
In other words, hover cars and bikes need to apply some of their energy constantly to fighting gravity; wheeled vehicles actually use gravity to apply their energy to lateral movement. Yes, these are simplifications of the physical laws, but they are functionally correct.
In point of fact, we do have a vehicle that performs the way you describe and it's been in service for many decades; it's called a helicopter. We don't like it in sci-fi for a simple reason; the rotors are too big meaning it can't get in tight and do all those dog-fight scenes. Also, they can really suck at those tight fast banking turns and rolls and the like that we like to see on films. But, it's actually a good fit for the rules you've given, and the point of it is that if we could make the rotors safer, smaller and make the craft more agile, we already would have. So the answer is probably no.
But, there is one technology that could make this issue go away; anti-gravity. Just like how in 1861 the Maxwell equations integrated electricity and magnetism into a single fundamental force and led to wholesale jumps in technical advancement that had not been possible before this, if we were to integrate gravity into EM as a fundamental force, and learn how to use EM energy to manipulate gravity directly, it could change the balance of energy use by allowing us to create localised fields of either counter-gravity or nullification fields of some kind. If you have that, all of a sudden you're not contsantly having to fight gravity, your vehicle sits inside a small bubble in which it just no longer exists. In such a situation, the only energy cost is moving the vehicle in 3 dimensions. Even just a localised effect (like the EM pads on the Matrix ships) could work insofar as they force a vehicle to hover by some form of gravitic repulsion. But, to do this right, you need gravity to be affectable via EM fields, and to do that we need to integrate the fundamental forces accordingly.
The caveat on this is of course that the moment your nullgrav field fails, you fall to earth and go splat. But, this has been the case for planes and helicopters for over a century now so we know how to handle that kind of problem.
In short though, if you're not happy with current helicopter design, then the integration of gravity into EM is the one defining piece of research you need to make such a device more energy efficient and easier to maintain / refuel in large quantities.