There defininitely are some restrictions: Segal constructs the Leray spectral sequence when the target (in this case X/G) is paracompact, and the stalks of the derived direct image can be identified with lim H^*(U_{hStab(x)};Q), where the limit is over Stab(x)-invariant neighbourhoods of x, so the vanishing result might use some property of X (local contractibility should be enough).