When I'm really overtired, I often find myself falling into a series of intense dreams or nightmares as soon as I fall asleep. My uneducated guess (only semi-supported by a brief internet search) is that it may be more likely you'll fall almost immediately into REM (rapid eye movement) sleep when very tired. Interestingly, dropping straight into REM is typical in narcolepsy. Which I'm pretty sure I don't have. Normal sleep patterns involve 1-2 hrs of non-REM sleep before going into the first REM stage. And non-REM sleep is thought to involve little or no dreaming, or perhaps less intense and less narratively based dreaming. It's still a pretty mysterious field of study.
When I realise it is happening, I am desperate to wake up and break the cycle. Sometimes it's hard to break out. Most recently I got to the last dream in the series and it was me lying in bed trying to turn the bedside light on but I couldn't make it work. Of course this wasn't real and after a struggle - probably just a moment in real time - I woke up properly and turned on the light. At this point I generally feel both scared and annoyed, because I know I'm overtired, I've probably made the effort to go to bed early, and now I'm losing the chance to catch up on my sleep and am not willing to slip back into sleep too soon.