Negative tension branes are a standard ingredient in many recently-proposed models for our universe. Although the presence of negative energy in a system sets off the alarm bells of relativists, the traditional reasons for this are often circumvented by placing the brane at a fixed point of a background orbifold space-time. In this talk I will argue that this construction may not be sufficient to guarantee stability within the context of general relativity. I will explicitly demonstrate a new instability in a toy model in 2+1 space-time dimensions, and present some preliminary results from a higher-dimensional study. I will conclude by speculating how certain string theory ideas may evade the constraints of GR.