We have performed magnetic susceptibility, heat capacity, neutron powder diffraction, and muon spin relaxation experiments to investigate the magnetic ground states of the 5d 2 double perovskites Sr2YReO6 and Sr2InReO6. We find that Sr2YReO6 is a spin glass, while Sr2InReO6 hosts a non-magnetic singlet state. By making detailed comparisons with other 5d2 double perovskites, we argue that a delicate interplay between spin-orbit coupling, non-cubic crystal fields, and exchange interactions plays a key role in the great variation of magnetic ground states observed for this family of materials.