Hello George,
every inbreath has a beginning, a middle and an end, every outbreath has a beginning, a middle and an end. Every beginning has a beginning, a middle and an end, every middle has a beginning, and so on.
If you fall asleep (and instantly awake afterwards), there may have been a (tiny) moment of unmindfulness, between observing your breath, falling asleep and recognizing it. Too much relaxation can e.g. lead to that state, as Metthew described.
In the book mindfulness of plain english, this moment of unmindfulness you may have experienced is called drowsiness/lethargy, which is quite the opposit of mindfulness.: "Note when it arises, how long it lasts, and when it passes away."
It seems to me, that you became mindful again, when you recognizing the passing away part (awakening). The tricky thing is to catch drowsiness when it begins. If you miss that part, it leads to more drowsiness and this often leads to sleep.
To me, when things like drowsiness happen, it is helpful to ask myself: What was the last thing i remember/what brought me out of the track (e.g. inbreath, or thinking about ..., or feeling very relaxed, ...) and how long was I away (e.g 20 minutes, 10 seconds,..).
Maybe it will be of any help to you too.
Greetings, Georg