In a world I'm creating, there is a sect of humans that hunts monsters in secret. All of them have magic "cores" installed on their person. These cores can store massive amounts of neutral biological matter (think enough to rebuild a person) as well as convert that mass into whatever tissue type needed and replace damaged cells (and repair DNA). The very obvious affect of these cores is practical immortality. But if they were fully (consciously and subconsciously) aware of the fact that they didn't have to hold back, could they be faster or stronger than a normal human?
Yes, they could be faster and stronger than a normal human, but not by a whole lot. The mind limits what it asks of the body to ensure that it does not injure the body, but a lot of this tendency gets conditioned out of elite athletes (weightlifters in particular) during the training process. From an article in Scientific American:
Vladimir Zatsiorsky, a professor of kinesiology at Penn State who has extensively studied the biomechanics of weightlifting, draws the distinction between the force that our muscles are able to theoretically apply, which he calls "absolute strength," and the maximum force that they can generate through the conscious exertion of will, which he calls "maximal strength." An ordinary person, he has found, can only summon about 65 percent of their absolute power in a training session, while a trained weightlifter can exceed 80 percent.
So you take a "no limits" weight lifter and a regular weight lifter of identical absolute strength. "No Limits" would be able to lift about 250 lbs while "Regular Lifter" would be able to manage a little over 200 lbs. No Limits would probably be in a lot of pain until the muscles they wrecked were repaired, while Regular Lifter would just have tired muscles.
Fatigue also appears to be another such limit. Your "no limits" people would be able to walk and run for ages, given that their "cores" would be constantly repairing the damage being slowly done to their bodies. They would also be able to forgo sleep without experiencing permanent brain damage. But the healing might take a while if the relative speeds of natural healing processes apply, however accelerated they may be. (Brains are really slow and hard to heal.) And they would probably still get dingy and hallucinate due to the toxins building up in their brains.
Going from ryepdx's answer which was great, I think someone who is going to have their body handle the damage done to it, can push it a lot harder than the 'No Limits' weight lifter. This will likely take months or years of training to be able to push themselves past the normal limits and even into limits that should kill them if they weren't 'special'.
Speed is going to have limits, these are going to be physical limitations that the body just isn't designed to do, however, they might take a lot less training to be able to sprint at max speed for much longer periods, and once you trained your body to not flinch when you hit something as hard as you can, because the damage will be repaired quickly you are going to be much more likely to perform amazing acrobatics and feats of strength that might break bones.
Basically once they have retrained fear responses and the bodies natural break limits many amazing things can be accomplished, but they are still going to be within what the human body can physically take.
Do they have Wolverine style healing? Then Wolverine is an excellent model for this (after he lost his adamantium skeleton).
One that you might not have considered - how do they deal with pain? That'd be a major limiter on the extra speed and strength.
Just because something's going to be fixed, doesn't mean its not going to REALLY hurt if you break it. A friend of mine wrote a short story that dealt with this quite cleverly. Just because you're invincible, doesn't mean you're going to throw yourself head on into danger with little regard to the consequences.
It could potentially be assumed that they'd be the peak of physical fitness because they would be able to continuously train without worrying about permanent injury so they could conceivably be pretty effective fighters, but not necessarily super-human.