There are some interesting results for simple faithful modules for finite groups for $n=1$ and 2.

In

M. Aschbacher and R. Guralnick, Some applications of the first cohomology group. J. Algebra 90 (1984), 446–460,

it is proved that, for a simple faithful module $M$ for a finite group $G$, we have $|H^1(G,M)| < |M|$.

In

Guralnick, R. M.; Kantor, W. M.; Kassabov, M.; Lubotzky, A. Presentations of finite simple groups: a quantitative approach. J. Amer. Math. Soc. 21 (2008), no. 3, 711–774,

it is proved that, for simple faithful modules $M$ for a finite group $G$ defined over a field, the dimension of $H^2(G,M)$ is at most 18.5 times the dimension of $M$.