Anon, you are correct: this shouldn't be happening to you. Also, 5-6 hours a night is not enough sleep for you. You should be getting 8-9 hours of sleep per night, so you are functioning on a big sleep deficit. You're not the only one not getting enough sleep nowadays, but for some other people the consequences may be years down the road, whereas you're experiencing them now.
The medical possibility that you need to have ruled out is called "narcolepsy" which means falling asleep for no reason, and falling asleep very rapidly, during the day. However, in narcolepsy, you wouldn't feel tired before falling asleep. You'd feel fine one second, and the next you'd be asleep. So it doesn't sound like that but you'd need to talk to a doctor about it.
Beyond getting enough sleep, another important factor could be what you are eating. You could definitely be reacting to what you eat. Either your blood sugar falling too rapidly, or a food allergy. Here's what to do to make sure your blood sugar stays at a good level all day long: http://j.mp/JPXm7L, and here's some information on food allergies: http://j.mp/ILRpI0.
The simplest approach would be to eliminate all sugar from your diet (fruit is OK) and make sure you eat a good breakfast with some protein in it. But reading those articles will give you a much better understanding.
Take care of yourself, and start getting enough sleep!
Dr. Daniel Heller