I will assume we are talking about an FPS here. I will say no, I dislike it, however, halo 1 had an interesting approach in which I accept it. A health bar, that if it goes out, you die, but you also had a regenerating shield over that. It made sense there.
Also, in the turok 2 multiplayer it made a bit of sense, and even fit with some of the classes, and was even put to interesting use by the fireborn, who after being damaged to near death, would be able to regenerate well past his initial starting health, provided you survive long enough for it to kick in and finish. (as with most of the characters that could regenerate).
So I guess I can say- It depends on how you include it. In call of duty I hate it, as its completely unrealistic... as with halo 2 and 3.
I like it when theres a consequence, and it forces you to think and allocate resources... Like in Halo 1... Or it 'adds' something to a character class, like in turok 2, then I will accept it. I would probably accept something like "stay out of combat for 20 seconds, then quickly recover to full health while being unable to do anything for 3 seconds" a valid way to put it, provided they could come up with an adiquate explination...
For a clearer picture of what I mean, refer to what others have already stated, my currently sleep deprived brain cannot think quite well.