For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E)\end{equation}For theories $h$ (such as ordinary homology or oriented bordism), are there results about when the sequence stabilizes (such as upper bounds on $r(p,q)$ such that $E^r_{p,q}=E^\infty_{p,q}$)?

What if we restrict to a class of nice spaces (such as classifying spaces $BG$ of groups)? For example, are there stability bounds for the Lyndon-Hochschild-Serre spectral sequence for group cohomology? In my limited experience, this sequence is often stable on the second page for finite abelian groups.