I seem to do quite a lot of wake up, look at clock, spend a little while awake, go to sleep again, repeat until getting up time, more or less like you describe, except maybe for the "fully" in "fully awake". This happens quite often.

Part of the problem is that in half-awake states I expect that my sense of the passage of time is even/far worse than normal, so as you say it's a very unquantifiable sleep.