I just read this post from Pamela with a great run-down of umbilical cord issues and what's actually going on.
I've heard so many times things like, "I would've died if my mom hadn't had me at a hospital, because my umbilical cord was wrapped around my neck and I couldn't breathe."
Ok... then how do babies breathe while they're in the womb? This mechanism doesn't just quit as soon as the head is born. Being born with the cord wrapped around the neck isn't an emergency, nor does it really compromise the baby in any way. More damage is done by cutting it prematurely, before the baby's body is born and it has a chance to start breathing on its own. As Pamela points out, you're effectively suffocating the baby by cutting the cord too soon.
I encourage everyone to read her post, it's excellent.