We numerically construct solutions to four dimensional General Relativity with negative cosmological constant, both with and without an electromagnetic field. Our results suggest that in both cases the space-time curvature can be made to grow without bound in a region visible to distant observers by imposing sufficiently violent boundary conditions on the metric or gauge field. In the electromagnetic case, this only happens at zero temperature, and we present a new numerical scheme capable of performing time evolution in this context. We argue that our results, at least in the electromagnetic case, violate the spirit of the Weak Cosmic Censorship Conjecture, so that this conjecture fails in 3+ 1 dimensional asymptotically Anti de-Sitter spaces. We then argue that if charged fields are included with a sufficiently large charge relative to their mass, cosmic censorship appears to be restored. The minimal charge agrees precisely with the bound given by the Weak Gravity Conjecture, suggesting an intriguing connection between this conjecture and cosmic censorship. More generally, we propose that "large" naked singularities, where the curvature becomes large over a large region of space, will be forbidden in any theory which can be completed into quantum gravity in the UV.