I would have assumed that, before sleep deprivation could actually kill you (directly, so not indirectly like, falling asleep while driving a car, which is frightening by itself), your body would basically just shut itself down.
I've also found that sometimes, when you REALLY think you've been awake for a given period, what really happens is your brain takes little "micronaps" every couple hours or so.
I remember that, when I was a teenager (after WWII, but before the Falkland war), my girlfriend gave me a book. Or rather, three books, each of them ruddy fat by themselves*. I started reading in Friday evening, and couldn't stop reading until I had finished them, by late Sunday afternoon. It FELT like I had been reading non-stop, but I *know* that, at some point, I reached a stage where I would doze off in my chair, only to wake up, say, half an hour later to continue reading.
I seem to remember that there is a sleeping schedule like that, which will actually sustain you for an extended period, getting by with something in between 2 and 3 hours sleep each 24 hours. It consisted of taking something like a 30 minute nap each 5 hours or so. The guy who tried this stopped it after a couple months, not because he was exhausted (he wasn't), but because it wrecked his social life (as well as his sex life, I imagine).
I'll see if I can find a link, and post it here.
*) Yes, you are right - it was "Lord of the Rings")