To do something surprising and seemingly impossible; to produce something in a way that has no obvious explanation, as if done by magic. A reference to the clichéd magician's trick of pulling a live rabbit out of a seemingly empty hat. The caterers really pulled a rabbit out of the hat by producing 100 cupcakes with just 20 minutes' notice. Unless someone can pull a rabbit out of their hat, I just don't see how we can fix the issue without starting the whole thing over again from scratch.