I did this a few times in my teens 'cause I was invincible then. :)
The one time it rained, I was using:
Clothing: 100% cotton shirt and pants, wool socks
Sleeping bag: 3-5 lb 30F-rated synthetic covered with cotton flannel
Bivy: 2 1.5 mil trash bags taped together with duct tape
Temp: around 35-40F
Although I was assured by others in my group that I'd be attacked and dragged away by wild animals, I managed to fend them off and survive the night. :) I awoke around 4 am to cold rain hitting my face. I simply covered my entire head and face with the sleeping bag and bivy, leaving a fist-sized opening in the side to breathe. The only issue I had was that sticks had punctured the bivy, and the bag and my pants were soaked from the calves down.
It was actually a fun experience. I'm tempted to try it again soon, but I'll probably use at least a small tarp over my head, and a breathable bivy.