Certain strongly disordered many-body quantum systems are incapable of reaching thermal equilibrium. The nature of this so-called many-body localized (MBL) phase has recently been an active area of research. The phenomenon can be understood through perturbative approximations, but rare regions with weak disorder (Griffiths regions) have the potential to bypass barriers to thermalization. I show that these effects do not destroy the MBL phase in one dimension, under a natural assumption on eigenvalue statistics. In higher dimensions, Griffiths effects may indeed restore thermalization on a very long time scale.