We show that a large class of supersymmetric compactifications, including all simply connected Calabi-Yau and G₂ manifolds, have classical configurations with negative energy density as seen from four dimensions. In fact, the energy density can be arbitrarily negative - it is unbounded from below. Nevertheless, positive energy theorems show that the total ADM energy remains positive. Physical consequences of the negative energy density include new thermal instabilities, and possible violations of cosmic censorship.