Rough answer : almost all small dims can appear, there are some restrictions to
large dims.

For example, considering 1 matrix all dims between 1 and n appear.
Taking centralizers of these all numbers of the form sum a_i^2 where a is a partition
of n appear.

In general, consider k-tuples of positive integers a and b such that their scalar product
a.b=n (a should be thought of as the Morita setting, b as the matrix-sizes of the semi-simple part 
of the subalgebra), then any number of the form

sum b_i^2 + subsum b_ib_j

is possible (here 'subsum' means just that one is allowed to take all these terms, or none or any selection)

There are some obvious restriction wrt large dimensions. For example, there cannot be an 8-dml
subalgebra of 3x3 matrices as its semi-simple part can be at most C x M_2(C) and so its dimension must be
smaller or equal to 7. For general n there cannot be subalgebras with dimensions between the dim of the largest parabolic
subgroup of GL(n) and n^2.