I actually saw that show, an alligators reaction to snapping their jaw shut comes from touch, So if you touch inside their mouth they automatically slam it shut and won't let go.
It was a hot day and the guy that was chomped had done this hundreds of times before, but as what happens when you work with animals, as he put his head in the mouth of the alligator a bead of sweat dropped of his head on to the alligators mouth or tongue and caused it to slam shut. They tried everything to get its jaw open including breaking huge lumps of timber.
The guy ended up with heaps of stitches in and around his head and a really bruised ego. Just goes to show that you can do this every day and one day you'll get bit. Bit like free handling vens I suppose.....