Can only speak for myself, but yes!! I often have at least that many.
Of course, I can't vouch for being normal . . . .
Actually, most teens and adults dream 4 to 6 times per night, as that is how often we usually enter REM sleep -- the Rapid Eye Movement phase of our normal sleep pattern. The length of REM periods varies across the night. It's shorter at first (5 - 10 minutes) and then a half hour or more later in the night.