Suppose we have a spectral sequence of algebras and know that it degenerates at some $E_r$, take for example the cohomology Leray Serre spectral sequence associated to some fibration $F\hookrightarrow E\rightarrow B$. Suppose we are working over a field so there is no extension problem and so $H^n(E)\cong\oplus_{r+s=n} E_{\infty}^{r,s}$. Under what conditions can we read off algebra structure of $H^*(E)$ from the $E_{\infty}$ page? (e.g. one very special case is when the $E_{\infty}$ page is free as an algebra.)

I think it is not possible in general, since we can have two different algebras with some filtrations such that the associated graded objects are isomorphic, but I would like to know if there are certain conditions under which one can compute the algebra structure.