I set a new record on my commute to work this morning: I fell twice because of the clips. I'm definitely new to it and my mistakes so far have been due to thoughtlessness. I have combination pedals that have platforms on one side and cleats on the other, and the first fall happened when I thought I was on the platforms when I was really clipped. The second fall was as Snorkel described: fully stopped, with one foot still clipped in, I reached for my water bottle and the bike's weight shifted the wrong way and dragged me down under it, right next to a light rail train full of people.
I'm not particularly easy to embarrass, and I quite like the clipless system while I'm in motion, but I'm starting to envy those people who claim they've never fallen in cleats!