That is the question I woke with today after having a series of dreams that each started with me dreaming that I woke up. In each dream, I would awaken, then go on to face a troubling situation. In some of the dreams, things turned out well, and it others, it didn't. But they were all exceptionally vivid, and made more so by the sensation of being wakened before each one began. I've come to the conclusion that I was dreaming...but only because the others who were involved in my dreams didn't share in my experience last night.
I had a shared dream, once, which also involved dreaming that I woke; when my sister and I were very young, we both dreamt one night that we woke up, went out into our backyard, and discovered that we could fly. Oddly enough, neither of us dreamt that we left our own yard; we just flew in circles around the lawn, about ten or fifteen feet in the air. Only the fact that we could not fly anymore convinced us that it had indeed been a dream; a shared dream, but a dream nonetheless. It can probably be explained by Peter Pan; but for a few hours, we were convinced that it had been real.
In my dreams last night, I faced each situation as it arose; and even when the results were not stellar, my courage was. Perhaps I am stronger than I suspect.