Almost all of the grad schools I applied to required GRE scores. I never got the impression that my scores mattered all that much, but perhaps I am projecting my own wishes, since I think that standardized tests are the easiest part of the application process to game.
I was successful at my grad school applications - I was accepted everywhere I applied, although one institution did offer only acceptance provisional on my completing an undergraduate course in a supporting science that I had no interest in and have never needed in work or grad school.
My GRE scores were... strange. I don't know what they predicted. I got in the upper 2% of all verbal GRE scores (stratospherically high compared to STEM applicants) and in the bottom 2% of all math GRE scores (pathetically low compared to STEM applicants). I had not bothered to study at all for the verbal sections (my sweetie pointed out that perhaps correcting the connotations/denotations in the vocab section of the test book was not the best use of my time), and then I worked my tail off on the math sections.
I do believe that my GRE scores are representative of my basic math/verbal skills - they correspond with how I've done in all of the standardized tests I've ever taken. But how applicable are basic math/verbal skills in a graduate program, one where you need to develop a thesis, work through the data, and determine the implications of what you've done? In my case, not relevant at all.
This post prompted by musings here and the comments.