Fault-based transition for crypto proof (a la Shoup) with big probability of fault - does it work?
My experience of such proofs is that often you're proving a much stronger statement: in any execution of Game i+1, either everything is distributed identically to Game i or else F must have happened. In other words, the conditional distribution of game i+1 conditioned on F not happening is identical to the distribution of Game i.

For example, one of Bellare's applications of this technique is the "poisoned point" principle: you choose the value of a hash function or similar at a particular point. As long as the adversary does not query the function at this particular point, the new game is identically distributed to the old one. If he does query this point, you abort.

Of course what you will argue next is that since you chose your poisoned point at random, the chance of the adversary querying this exact point is negligible. But the actual game-hop doesn't depend on the probability of F happening, it only says either all is well or else F must have happened.

Failure events with non-negligible probabilities happen frequently in hybrid arguments. They tend to make for very lossy reductions of course but the principle stays the same, the hop itself is Pr[S_{i+1} = 1 | not F] = Pr[S_i = 1] and then you compute the probability of F in a separate step to get your "concrete security" constants.