I think I've had more than a couple of experiences like you described, about 2 years ago. It was a Polish language lesson, with a boring teacher, at 7:00 (essentially requiring me to get up at 4:30, I live far away from the school) and I'm not a morning person. Not to mention I was much more active in various FS projects back then, so I often stayed up late to catch people from different time zones on IRC. When I usually didn't remember a thing from those (in most cases I took notes somehow, but if I didn't, it was awkward asking my classmates for the info I should have), few times I was more awake than usual (the usual being, IIRC, passing out into black for a short time, not long enough to get dreams) I remember it being pretty much like you described. It was like the real life was running fullscreen in the background, with something else (in most cases more interesting) running windowed on top of it. Sometimes it was difficult to tell the dream from reality.
It's a bit of a fuzzy memory, because by the end the 1st year at high school I felt like I'd end up hospitalized if this continued for much longer (well, that, and there was this time I almost walked in front of a car when crossing the street on my way to school. I started skipping that blasted lesson after that), so my parents made arrangements so I won't have to get up this early anymore. I don't know about the others (it's different for every person, some of my classmates didn't had problems getting up that early), but this kind of "sleep" hardly provided me any rest, or at least not enough of it to make me feel comfortable for the rest of the day.