Unless I'm missing something, as long as for each target group H_k, the part of the spectral sequence calculating E_\infty H_k, has no differentials in or out after the nth page (which can depend on k) then the spectral sequence will strongly converge under these conditions.

I second Drew's recommendation. The paper is complex, but wonderful. The key ingredients are: A spectral sequence should be associated to a good filtration: the intersection/union of the filtration that should be 0 is very much 0 and conditional convergence gives this by definition. You should also be sure that the respective union/intersection of your filtration actually calculates what you are looking for. In practice, checking these conditions is easy. Under these conditions the only thing to check is that

Hi Greg, perhaps you mean $X^H$ is contractible for all non-trivial subgroups $H\subset G$ ($G_+$ is cofibrant, but is not contractible for all proper subgroups unless $G$ is trivial).

@AaronMazel-Gee: Sorry, that was less than clear because I rushed the answer. I filled out the answer a bit more. Hopefully, it is more understandable now.