Aaron is right, this question would benefit from narrowing down. However, the fact that you encounter the same issue in multiple disciplines points to some issues in the way you approach what you're doing, and a few of your phrases offer clues that you might want to shift your assumptions and philosophy in ways that can impact your body.
First of all, you mention "increasing the speed of a small repetitive movement." Sure, we often have to do that (a drummer practicing rolls, a guitarist practicing tremolo, or simply increasing the tempo of scales). But I would encourage you not to make speed the only, or even the primary, measurement of progress. There are limits to our ability, and we have to increase them gradually. As a runner, if I try to train too hard too quickly, I'll injure myself. As a musician, if I set myself to play a scale over and over a little faster each time, I will encounter my limit, and must accept that limit today, be willing to move it only slightly each day, and make peace with the fact that at some point I will encounter my permanent limit. Also, there are more ways to grow as a musician than by doing something faster. Besides working on expressivity, creativity, etc., even if we're just talking about technique, sometimes it's important to slow down to work on precision, nuance, and accuracy.
Next, you mention that you've tried addressing this tension with "mental focus" and "journeying to the edge of your body's ability." You know best what you're actually doing, but these phrases suggest to me "trying harder"; increasing physical exertion to find your "body's ability" and increasing mental exertion by "focusing" or concentrating harder. This might explain why you've had negative results. There is almost (almost) never a situation in musical practice where the answer is "increase exertion." Try to find places where you can address the physical challenge by working less rather than more.
First of all, in many of these situations, you might find that you're engaging muscles that are not actually contributing to the motion. As a violinist, this reminds me of working on tremolo (I imagine it presents a similar challenge for guitar/mandolin—I'm talking about picking one note rapidly). As I move my arm backward and forward, I wind up just tensing both sets of muscles and staying in stasis. Think of the muscles you would engage if someone were trying to move your arm all over the place and you wanted to resist them: You'd have to engage opposing sets; biceps vs triceps, etc. You'd "lock up" your arm and wind up going nowhere. To get out of that pattern, first ask yourself if you're engaging any muscles that aren't part of the process. What are the smallest muscles you can accomplish the movement with? Do you even have to move your arm, or will your fingers do? Can you move at the wrist? If someone did come along and bump your arm while you're executing this motion, would your wrist bend, or are you using opposing forearm muscles to lock it in place? It can be helpful to use a mirror to actually observe your body, or to get a friend to actually try moving your body. Also, remember that as you increase the speed of a movement you can decrease its size. If a violinist plays a scale at 60 bpm their bow travels a certain number of inches per note; at 120 bpm it ought to be able to travel half the distance. As you increase speed, try to shrink your motion.
Next, as has been mentioned, the mental part of the process is at least 50%, more like 80%. Here as well, try to "work less" rather than "work harder." Any yogi will tell you that physical flexibility and relaxation is impossible without mental relaxation. Instead of increasing "mental focus," thinking fiercely and focusing your awareness on one muscle group, broaden your mental focus. Relax your mind as well as your body. Stay aware of those irrelevant muscle groups. As you practice your picking, what are your legs doing? Is there tension in your thighs? Your torso? Your back? As you increase the speed of an exercise, try to reduce your excitement rather than increase it; tell yourself "it's easier" and you might find yourself changing the physical motion in ways that are in fact easier.
And finally you say "I've practiced enough." I'm sure you have, and it's important to be able to say that long before your mind or body have reached their limits. Know when you've practiced enough for one day, and when you've simply practiced that particular movement or passage enough and should try something different. I used to walk through the hallways of practice rooms and hear a pianist practicing one measure, over and over, 10 times, 20 times, 30. Then I'd leave for dinner and come back 2 hours later and they're still on the same measure—what is that, the hundredth time? 200th? They're accomplishing nothing but injury. At some point, it's time to practice smarter, not harder. If you missed a note or messed something up, don't just try again until you've asked yourself why and what you can change.