When does the Grothendieck spectral sequence converge?

Question

I am trying to understand spectral sequences in algebraic geometry. One has the Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$, and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories with enough injectives. This spectral sequence converges under mild conditions, such as $\mathcal F$ maps injectives to $\mathcal G$-acyclics. Here is my question - suppose one has derived categories instead of abelian categories. What is the analogue of the above condition for convergence? Can one say something in terms of convergence of the Hypercohomology spectral sequences of $\mathcal F$ and $\mathcal G$?