Seems very unlikely to me. Quite apart from the willpower required to inflict such ongoing pain over a period of time, you're going to find that the loss of blood will be fairly low (in comparison to the total amount in your system) and will tend to coagulate as you go along. Even hitting veins is no assurance of success, whilst arteries such as the Femoral or common Carotid are relatively deep and protected by muscle.
Sorry if I sound negative, however I just can't see this one being a goer.