Yes, it happened once when I tried cutting her nails while she's awake. So what I did was I applied pressure until the bleeding slows down. Then I covered it with gauze. Since then, I always cut her nails when she's asleep.
It happened with me once. Though, fortunately it got just the minutest prick kind of a thing but I freaked out more than my daughter. I have been super cautious since that one time whenever I sit to cut her nails.