I have never heard of this happening to someone, but I believe it's possible. For those of you who don't know, you go through hour-and-a-half sleep cycles while you sleep. So if you sleep for a healthy nine hours, you go through six cycles. During these cycles, you go from light sleep, to medium sleep, to deep sleep, to R.E.M. sleep, which is the scary portion of sleep. Then you go backwards again...
Often times during the light sleep stage you wake up, but not always. Also, dreams occur in all phases, but they differ. In lighter sleep, your body is not completely paralyzed yet, and your dreams are more like thoughts than stories. If you think your are about to get ran over by a giant marshmallow, you will jerk awake in light sleep. However, you typically cannot wake up from deep/R.E.M. sleep very easily.
What is interesting in dreams during the various stages is that light sleep dreams are much swifter, and do not activate the brain in as many areas or as much in general. During R.E.M. sleep, (named after rapid eye movement) time in dreams slows down to match real time, and your brains fires up just like it does in real life. R.E.M. dreams, as stated before, are scary. They are frighteningly realistic and are more likely to be negative and play on your fears and desires than other dreams. Your body is completely paralyzed at that point, so you typically cannot wake up unless you get shot in the dream or fall off a cliff, etc... Not all R.E.M. dreams are awful, but they are the ones that are designed to prepare us for the real world, which is often not a nice place.
The sleep cycles have more deep sleep during the late night hours, from 1:00 to 6:00 A.M., so sleeping too much during those periods and not enough before midnight can cause an individual to get too much R.E.M. sleep, and actually contribute to depression and a bleak outlook on life.
When you do wake up in the morning after all that complicated stuff, most people say they either don't remember their dreams or they didn't have any. In fact, as long as you have slept, you have dreamt, unless you have a severe and rare sleeping disorder. Humans typically only remember 10% of their dreams the morning of, and they can be quickly forgotten throughout the day because they have the label of "useless information" in your mind. The mind throws out many dream memories as quickly as possible because many of them are pointless and unrelated to the real world, but if you really want to remember dreams, don't move around a whole bunch as soon as you wake up. Stay put and try to recall what you dreamt about. Moving a major muscle, such as the bicep in your arm, clears your memory of the night, unless you have recited it and thought about it already.
On rare occasion, an individual will be unable to move when they wake up because the paralysis caused by deep sleep does not ware off when one awakes. This means you will have to sit there until it does, which could take up to a half hour, or just a few minutes. If this happens to you, you can try moving small muscles and moving onto bigger ones, such as your eyelids or fingers.
Another interesting thing that can happen when you wake up is... not waking up. As I stated in an earlier post, lucid dreams often have the effect of overexciting individuals and causing them to wake up for real, or maybe just waking up in the dream and going through a daily routine in the dream. So maybe next time you wake up in the morning, try to fly. It just might work.