I've done this to Isaac twice. Once was the first time I used the nail clippers made especially for dogs, which ironically enough are supposed to prevent you from cutting the nail too short. The second time was yesterday.
The first time I did it, Isaac let out this yelp like I'd chopped off a toe. The nail bled like crazy. I mean, there were big red drops of blood all over the carpet and everything. I felt terrible and was really scared to cut his nails again. The reason I did cut them again, and later that very same day (I needed to finish the job because I stopped when I cut the nail too short), was because I didn't want Isaac to become scared of having his nails cut. I thought finishing the job later the same day would help. I wanted to do it without acting afraid at all (which was not easy for me because I was afraid) and I wanted him to see that he could get his nails cut without it hurting him.
Yesterday when it happened, I still felt bad, but I'm not scared of cutting his nails again. I didn't panic as much this time. I knew what to do.
So here's a handy little trick. I keep a glass jar (used to hold salsa) with cornstarch in it. To stop the bleeding, just stick the dog's toes into the cornstarch. In less than a minute, the bleeding was stopped. I can't remember who told me that trick. It might have been Isaac's trainer. I'm not sure why or how it works, but it does.