In homotopy theory there is the following informal idea:

> The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far from the actual answer.

Are there any similar uncertainty principles in other areas of mathematics? To be more specific: give examples of some abelian groups (or other "abelian-ish" objects) that some people find interesting, that are very hard to compute and that are the $E_{\infty}$-page of some spectral sequence whose $E_2$-page is relatively easy to compute?

This is, to some extent, a question of format but I believe that it does have non-zero mathematical content (probably not everything can be reduced to a spectral sequence). I just have a S.S. phobia so I am trying to understand in what ways they can arise outside of homotopy theory.