So let's say that a single one of these threads, perfectly constructed, has a strength of 1. So if you have three of them tied off to the same object, you should expect to get a strength of 3.
But threads aren't perfect. At some point, every thread is stronger than other parts, and weaker than other parts. If you have three separate threads lifting a load of 3, then they can fail one-by-one at those weak spots.
But if you roll them together, they bind. So in most cases, the weak spot on one thread is beside a non-weak spot on another. It may even be beside a strong spot on another, so that the total strength along the length is roughly constant.
So if you have 1000 threads woven in a rope, in the worst case you might have a spot with a strength of, say, 950. Whereas if you have 1000 separate threads, you have strength 999, then 998, and so on.
In fact, there are some ways to make ropes where they actually get stronger as individual threads fail. The trick is that they are woven so that normally bear the loads on an angle, but as the threads fail the remaining ones straighten out and provide more strength. IIRC you have to cut something like 20% of the threads before they begin to weaken. Unfortunately, I can't recall the name of them, although I learned about them as part of a proposal to make a high-altitude balloon-borne quasi-permanent platform.