Many observations suggest that much of the matter of the universe is nonbaryonic. Recently, the DAMA NaI dark matter direct detection experiment reported an annual modulation in their event rate consistent with a WIMP relic. However, the Cryogenic Dark Matter Search (CDMS) Ge experiment excludes most of the region preferred by DAMA. We demonstrate that if the dark matter can only scatter by making a transition to a slightly heavier state (∆m~100 keV), the experiments are no longer in conflict. Moreover, differences in the energy spectrum of nuclear recoil events could distinguish such a scenario from the standard WIMP scenario. Finally, we discuss the sneutrino as a candidate for inelastic dark matter in supersymmetric theories.