A rather less naive bound on the sizes of cohomology than the one I sketched in a comment to the question follows from the famous result of Venkov that if $G$ is a finite subgroup of $U(n)$, then $H^*(G,\mathbb F_p)$ has rational Hilbert series of the form $\frac{r(t)}{\prod_{i=1}^n(1-t^{2i})}$ with $r\in\mathbb Z[t]$, and such that the pole at $1$ has order equal to the $p$-rank of $G$. This gives very good bounds, although for each prime at a time.

To deal with the case of more general coefficients, there is the whole theory of complexity of modules.