It depends a lot on your own personal sleep needs. I've gone for ~35 hours with only a short nap on the floor somewhere in the wee hours and it was fine. 24 hours with no nap is usually fine, too, although I can benefit from a short nap. It helps to be very well rested when you start, obviously. Sometimes it makes a difference what time of day you start; it's easier to stay awake when there's daylight than in the dark.
But people's sleep needs really vary. Personally, I really need 8 hours a night on normal days. But I have a friend who sleeps 4-5 hours a night tops, and has always been that way. At least one of his kids is like that, too. He came with us on a brevet once, and complained about having to just stand around while the rest of us took naps on the ground. He just wasn't sleepy.
The most sleep-deprived I've ever been was probably a 1267k I did where I kept getting lost and wound up with an hour and a half the first night and nothing longer than 20 minutes or so after that. What with going over mileage by 180 km and all, I finished in 98 hours. I am pretty sure that some of my navigational difficulties later on in the ride were probably related to getting stupider and stupider at reading maps and cuesheets. I didn't hallucinate, but at one point I got separated from the guy I'd been riding with and later neither of us could remember when or how. I do remember looking at the map and having a hard time figuring out what all those squiggly lines meant. I was nodding off constantly that last night, and was at a point where even sitting up and slapping myself in the face with both hands didn't work and just felt like too much work.
But generally if I can get a couple hours of sleep per night on a longer brevet, I'm sleepy at times but able to function without nodding off on the bike.
I've also found that getting chilly makes me feel sleepier before it makes me feel cold, and not eating enough makes me feel sleepy before it makes me feel bonky. I also tend to get sleepy on descents, between the colder wind and the drop in heart rate after the climb.
Of course, YMMV.