There are two major limits to this technique which you would need to overcome.
The first is that evolution is not dumb. Well, it is dumb, as in it has no goals, but it has done a very good job of making a body structure which keeps you alive. There are only a small number of places where there are key unguarded "cords." Typically the arteries are deeply buried. Even veins are reasonably protected If you ever wonder why angioplasty, a heart surgery, goes in through the groin, it's because the femoral artery is unusually accessible at that point. You only have to push around a few layers of muscle to get access to it. Thus you would need your martial artist to have the skill and strength to cut all the way to the cord before cutting it.
There are a handful of tendons that might be easy targets. The Achilles tendon appears in some sword martial arts as a weak point, if you can find a chance to strike such a small target meaningfully. Also the tendons in the wrist and the back of the knee are rather exposed.
The other issue is that cutting a cord actually calls for cooperation on the part of the opponent. Try this: get a length of string and a pair of scissors. Hold the string up on one side and try to cut the string below it. It's nearly impossible unless you have some good sewing scissors. You need tension on the string to really cut it. Otherwise it gets out of the way. Now repeat this process with a knife, rather than scissors. It's even harder because we don't have two opposing pieces of metal. You're trying to do this with a finger: go get an ice pick and try to cut that string!
Fortunately, opponents will often tense such cords for you. If there is stress, there are tense cords. Part of your technique would need to be convincing your opponent to fight you in positions of poor mechanical advantage. Offer a "vulnerability" for them to strike. All they have to do is over power you. However, in the process of overpowering you, they'll tense their cords, and you'll have something to attack.
If you look at the best martial artists around, you'll find that they don't leave cords tense for a long period of time. Doing so is very wasteful from an energy management perspective. Only the weak martial artists make the mistake of leaving things tense longer than they have to. Even in sports like boxing and MMA, which don't have the word "art" in their title show this skill, so you'd find it very difficult to take on a skilled fighter, whether they practiced a "martial art" or a "fighting sport."