It's not like it hasn't happened... though if the same person does it often enough, there is a very real possibility of being charged with 911 abuse at the very least. Then you get a free ride that you can't leave whenever you feel like it.
This is not an infrequent occurrence at some of the northern services here that serve First Nations Reserves. They are often remote and extremely economically depressed with the hospitals being in the largest local urban centre. The patients don't get out part way, but they do leave the ED AMA shortly after arriving. I don't believe it's a daily occurrence but every medic I've worked with that's come down south from working up there has done it once or twice.