Here's an interesting statement from an article on terminal ballistics:
The first hit is the most important, because endorphins that are released into the body as a result of serious injury cause the constriction of the blood vessels, reduce or eliminate most pain, and condition the body to operate with minimal oxygen in the blood. If the first hit is not immediately lethal, then subsequent hits will often be less effective in quickly dropping the target, even if they are lethal wounds. Few people realize this, but it is well documented. It is the same mechanism which makes a startled or alert deer harder to kill than one which is completely surprised, because fear also triggers the release of endorphins.
http://www.rathcoombe.net/sci-tech/ballistics/mechanics.html
From what I've learned from my introductory psychology class, I can say that the endophin release part is true and personally I think it makes sense about the next few shots not having as much of an effect. (Note that it does not say it has
no effect, just less of an effect)
For me, where I hunt, it's so thick that I have to make the first shot count because a follow-up shot is quite rare and I have never taken one. However, I do have four more rounds ready to go, if a second or third shot were to present itself.