We prove that the necessary condition for a solid to be also a superfluid is to have zero-point vacancies, or interstitial atoms, or both, as an integral part of the ground state. As a consequence, in the absence of symmetry between vacancies and interstitials, superfluidity has a zero probability to occur in commensurate solids which break continuous translation symmetry. We discuss recent 4He experiments by Kim and Chan in the context of this theorem, question its bulk supersolid interpretation, and offer an alternative explanation in terms of superfluid interfaces.