When thinking of cohomology as describing a defect to a functor being exact, it has to be expected that the first few $H^i$ appear more often. But there are of course higher coholomology groups and they appear naturally.

However over local and global fields, they can be described in terms of $H^0$ and $H^1$ usually. First over a non-archimedean local field $k$, $H^2(k, E[m])$ is dual to $H^0(k,E[m])=E(k)[m]$ and $H^i(k,E[m])=0$ for $i>2$ when $m$ is coprime to the characteristic, and $H^2(k,E)$ vanishes. That is local Tate duality. 

For a global field $k$, the kernel of $H^2(k, E[m])$ to all its local versions is dual to the kernel of $H^1(k,E[m])$ to all its local versions. Again $H^i(k,E[m])$ vanishes for $i>2$ and odd $m$ coprime to the characteristic. This is part of the long exact sequence of Poitou-Tate. 

etc. There is much more to be told. But in fact most of the above is not specific to elliptic curves and a book like "Cohomology of number fields" will describe this in all details.