Chronically, no it isn't normal. Ocassionally, it's not uncommon.

The dreaming of waking up is actually a trademark of Theta sleep, which isn't so much a dream as it is an active visualization. Theta sleep is the stage when the body hasn't fully given itself over to dreaming. Your mind is nearly there, and your waking body is more or less straddling the fence. You think you're in a dream because there is juuuust enough influence from your subconscious to effect what you see, but the waking world's influence pretty much forces the visualization to mimic reality. The repeated waking up is basically when your waking mind tries to force consciousness....and fails. So you fall back onto the fence, only to have another "Hey, I'm dreaming" moment and another chance for your waking mind to force consciousness again.