Can you die of being tired?

If there are misspelled words, utter nonsense, or any kind of emotional outbursts in this post, it is because I am inexplicably, incredibly, no-joking-about-it exhausted. If Josh were writing this, it would be that same kind of tired except doubled.
On average we both have been getting around five hours of sleep a night since school/work picked up. I can't put my finger on it exactly, but I think we have reached the point where, if we could stop and sleep, we wouldn't because we are too sleepy to sleep. Does this make sense? It just occurred to me that it will only get worse when we decide to have a baby. Gasp.
One of these too crazy tired to sleep nights happened last night. I was examining my toes, going cross eyed with the effort and lack of shut eye, when I asked Josh, "Could someone die of tiredness?" (Long pause while I feel my sleep devil prodding me awake with his pitch fork.) Josh went into a deep, sleepless stupor considering my question. I continued, "I mean, could your organs just get so tired they would cease to function? Would your heart and brain want a break so badly they would just take one without your permission?"
After we walked around Wal-Mart at midnight looking for a specific brand of granola bars for my nutrition class, Josh came to the sound conclusion that someone would pass out before their body would stop working.
I think I like this conclusion as much as he does. I think we are going to try and get some sleep...another night...maybe...if possible (please?).