This is a funny question, which has arisen as a critical component now and then. Yes, if we insisted that the subalgebra be *semi-simple*, then Schur's lemma gives the result, as in a comment. Perhaps some readers overlooked the possibility of getting $[n^2/4]+1$ (for the matrix case) by taking scalars together with upper-right-corner blocks, with everything else $0$.

The best reasonably-provable result I know was proven by D. Kazhdan as a part of J./I. Bernstein's proof of the admissibility of supercuspidal repns of p-adic reductive groups, and (for the matrix=split case) says that the dimension for a subalgebra with $\ell$ non-scalar generators is bounded by $(n^2)^{1-2^{-\ell}}$.

Since it's not obvious how to exceed $n^2/4$, this bound seems needlessly weak, but we can suspect that Kazhdan would have given a sharper bound if a simple one really held, at least as a meta-argument for no sharper bound holding. I've not tried to construct an algebra exceeding the $n^2/4$, although I have tried (and failed) to improve that bound. The argument is surprisingly non-trivial. A very sketchy version appears in Bernstein's original paper (in J. Fun. An. and Applications), a somewhat fuller version (revised after some comments from D. Renard) in my essay http://www.math.umn.edu/~garrett/m/v/proving_admissibility.pdf, and as an appendix in David Renard's relatively recent book on repn theory of p-adic groups.