Let $X$ be a manifold such that $dim(X)=n$. It is well-know that if $\mathcal{F}$ is a coherent sheaf $H^m(X,\mathcal{F})=0$ for all $m >n$ (where I denote with $H(-)$ Cech cohomology). But is there a sheaf such that $H^m(X,\mathcal{F}) \ne 0$ for $m>n$? 



Vanishing Cech cohomology
I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by [this answer](https://mathoverflow.net/a/43405/75), the cohomology of any sheaf vanishes in degrees above $n$ on any topological $n$-manifold. Essentially, it can be shown that any open cover of an $n$-manifold admits a refinement for which all $(n+2)$-fold intersections are empty, so the Cech cohomology automatically vanishes above $n$.


