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Just a little something to add to this - the 4x4x4x4 4D board is 16x16 when
laid out in 2D to actually play on. Yet, because the movement rules are
written for a 4x4x4x4, the king, or any other piece, *cannot* move more
than 3 squares in any one 'direction'. 
I do agree that 'size' is related to dimensionality. A 4x4x4x4 4D board
is the smallest really useful 4D board. A 3x3x3x3 needs gimmicks to work
well, or at all, and isn't big enough for pieces to have any scope for
their moves. How does a knight move on a 3x3x3x3? Put it in the middle,
and it has no moves at all. The same sorts or arguments can be made for
3D, where 4x4x4 is tiny. I would argue that an 8x8 in 2D is also a small
board, though, so you'd want to take this discussion with a grain of
salt.
Possibly a better criteria would be what people will actually play. Given
recent history, the 12x16 is the very largest that most people will try
now. For those of us who like the larger sizes, the trick is to come up
with enough good games to bring the others along.
This leads us to the question: What makes a large game interesting? Just
what can you do on a large board that you can't do on a small one?