How can a five-dimensional black hole 'break' general relativity? By the laws of general relativity and the existence of black holes, what makes our universe to be just 4 dimensional but not 5 or more because a 5-d black hole can violate theory of relativity?
Reference: End Point of Black Ring Instabilities and the Weak Cosmic Censorship Conjecture
Does this mean the "Theory of Relativity" is wrong?
 A: Nothing about the type of system described in the paper breaks general relativity in the sense that it causes an internal inconsistency. However general relativity is just a mathematical model, and while the mathematical model is fine what is not so obvious is that it correctly describes the real world. Ultimately we can only judge this by experiment, which is hard because it's hard to find systems extreme enough for GR effects to dominate that are easy to experiment on. The observations we have been able to make have so far supported GR in all respects (add the recent LIGO observation to this evidence!).
While experimental evidence is hard to come by, there could be theoretical evidence. For example if GR could be shown to make a prediction that was physically absurd this would be evidence that GR is not a good description of the real world.
One of the features of GR is that it predicts curvature singularities i.e. points in spacetime where the curvature is undefined. For why this is a problem see my answer to if singularities can be observed from the rest of spacetime, causality may break down. However as far as we know these singularities are always hidden behind an event horizon and this makes them harmless. This principle has been raised to the standard of a conjecture - the cosmic censorship hypothesis.
What the paper claims is evidence that in the 5D system they describe a form of the cosmic censorship hypothesis called the weak cosmic censorship hypothesis may be violated. I don't know enough about the subject to comment on their working, but let's be charitable and assume that their conclusions are correct. Would this be evidence that GR is not a good description of the real world?
Well firstly there's the obvious point that they are working in 5D. GR claims to represent our four dimensional universe, so the fact that weird behaviour crops up in other dimensions shouldn't trouble us.
Secondly it isn't obvious that a GR prediction of naked singularities would actually be a problem. Pretty much every physicist I know expects that singularities don't actually exist and that some future theory of quantum gravity will replace them with a non-singular geometry. When GR predicts a singularity that just means we've entered a regime where GR needs to be extended to include quantum effects.
And even if naked singularities really did exist, would this be evidence for anything other than that the universe is stranger than we though? Mankind has been discovering that the universe is stranger they we thought since we first climbed down from the trees, and naked singularities would be just one more step in this journey.
