In principle doable but tricky. 16 x 16 x 16 is 4,096 individual RGB LEDs (or 12,288 individual LEDs).
Power would be tough. Using a modest figure of 5mA per LED that's 61 amps of power needed. Obviously you wouldn't light all LEDs simultaneously but it gives an idea of the difficulty involved unless you substantially limit the output.
The other issue is cost. Doing some very rough calculations you'd be looking at a retail price well north of $1,000 which is probably not something that would interest many people. :-(
A final issue is visibility. LED cubes tend to be constructed from through hole LEDs because their lenses emit in all directions. The wiring tends to be the leads of the LEDs themselves soldered together (so very thin and not obstructive to viewing). PCB mount RGB LEDs point in one direction meaning that the cube would only be optimally visible from a single direction.
It's possible that something could be done using "dumb" RGB LEDs (the APA102s are relatively expensive) but that raises the complexity of implementation a lot!