h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

It wasn't done 2,000 times that I've ever heard of... but you could. The difference between paper and steel is that when heated and folded and flattened, steel deforms; it doesn't run into the problems paper does.
I can't swear to it, since I'm just beginning to learn the metallurgical part of swords and their construction, but I suspect that the reason you'd want to fold the steel of the sword so many times is to help make the steel of the blade more uniform. I know that in the west, there was often a problem with steel being too soft in some spots and too hard in others, up until relatively late.

Well, I should correct this, having found out the reason, and actually remembering it this time.
A folded katana is done thousands of times, and this is for the same reason that pattern-weld or "Damascus" steel, as it's known, does it. Steel bends, and welds to itself. You take a layer of hard metal and a layer of soft metal, put them together, heat them, and fold them. This has the effect of giving you a blade that will hold an excellent edge, but not break easily.

It is also believed that the best katanas used alternating layers of annealed and quenched steel, The two types work in tandem to give a better result than just using tempered steel...

My understanding had been that Watered or "Damascus" steel was actually a different thing than pattern-welded. In pattern welding , you use two types of metal , but watered/Damascus/katana steel is folded many times( not 2,000 times. 2,000 is the number of layers, there are fewer folds than that. Also, only relly good swordsmiths got 2,000 layers.