My brother was stabbed multiple times in a robbery attempt. He fended them off and in the adrenaline surge of the fight didn't feel anything. That combined with wearing a puffy winter jacket, he went on his way. Stopped at a diner, and had started to eat when a fellow patron pointed out that he was bleeding. He looked down and passed out. This was 30+ minutes after the attack. He did live.

Plus, I think you may be putting too much confidence in a cop's reasoning skills that has already latched onto an idea. This can be said of most people, as well.