A Bill Murray story is when you create an outlandish (yet plausable) story that involves you witnessing Bill Murray doing something totally unusual; often followed by him walking up you and whispering, "No one will ever believe you" and walking away.
My freshman year of college, I was hanging out with some friends playing Halo 2 in my dorm. Sure enough, I hear a strange meowing sound coming from outside my window. As I open the curtains and look out, there was Bill Murray, clinging to a branch about 10 feet in the air, meowing at a kitten stuck a little higher up the tree. He turned around and looked at me, and mouthed 'No one will ever believe you' before climbing out of sight.