Lots of people do this. I recall reading that Gary Strydom used to eat chicken breasts every four hours, 24 hours a day, 7 days a week. I'd be gagging on them after a few days.
I figure if you wake up, it will help to down some protein, but I wouldn't set my alarm to get up to do it.
I do think it is a good idea to have a protein shake before bed. I use caseinate protein in mine since it digests slowly. I also mix in peanut butter or flax oil with the powder and milk since this is supposed to further slow digestion and keep aminos available most of the night.