The Berge Conjecture attempts to classify knots in S3 which admit a (non-trivial) lens space surgery. After providing some background for this problem, we will discuss why the natural generalization of this conjecture to knots in lens spaces which admit a second lens space surgery fails. Following this discussion, we will see how to make some sense of these counter-examples. This is joint work with Ken Baker, Nathan Dunfield, and Joan Licata.