The Principle of Alternative Possibilities is the intuitive idea that someone is morally responsible for an action only if she could have done otherwise. Harry Frankfurt has famously presented putative counterexamples to this intuitive principle. In this paper, I formulate a simple version of the Principle of Alternative Possibilities that invokes a course-grained notion of actions. After warming up with a Frankfurt-Style Counterexample to this principle, I introduce a new kind of counterexample based on the possibility of time travel. At (...) the end of the paper, I formulate a more sophisticated version of the Principle of Alternative Possibilities that invokes a certain fine grained notion of actions. I then explain how this new kind of counterexample can be augmented to show that even the more sophisticated principle is false. (shrink)