Yes, sometimes it would be a continuation of the dream, as in it picked up where I left off when I woke up the first time. More concerning than that is when you have that same dream multiple times, including waking up in the same spot, going back to sleep and continuing as it did before. Very weird when you know what is coming next and you have to try and fall asleep again. One step above these dreams are the dreams where you know you are dreaming, and you try to wake up, only to discover shortly after that you are in another dream. I once had a bit of a freak out when I kept waking up in a dozen or so dreams nested in each other. When I finally woke up for real, I was drenched in sweat and crying, it took more than a couple minutes before I was finally convinced that I was actually awake and not just in another dream.