Let's assume we have a 4x4x4 Rubik's cube. With this cube we can imitate a 2x2x2 by considering all 2x2x2 corner cubes as 1 block. Similarly, we can solve it like a 3x3x3 by first solving the 2x2 centers, and then solving all edge pieces, and then consider them as one. This might give some parity problems in the end, but let's ignore them.
With a 3x3x3 however, we cannot imitate a 2x2x2, and with a 5x5x5, imitating 2x2x2 and 4x4x4 is not possible.
What is the general rule for this, i.e. given an $n\times n\times n$-cube, for which $m<n$ can we imitate an $m\times m\times m$-cube on this cube (ignoring possible parity-problems we might run in to)?
It seems that for odd $n$, we can only imitate odd $m<n$. But for even $n$, the rule seems to be stranger.
With an $n\times n\times n$ cube, we can easily emulate an $(n-2)\times(n-2)\times(n-2)$ cube by always moving the two outermost planes together. By recursion, we can emulate any smaller cube with the same parity.
If $n$ is even, we also can simulate an $(n-1)\times(n-1)\times(n-1)$ cube by always keeping the two middle planes together. Therefore for even $n$, we can emulate every smaller cube.
Obviously, with an odd-$n$ cube, you cannot emulate an even-$n$ cube because you cannot turn half of the planes.
So in summary, with an even-$n$ cube you can emulate every smaller cube, while with an odd-$n$ cube you can only emulate smaller odd-$n$ cubes.