Four Theories of the GRE's evaluative significance:
1. The Primacy of Quantitative Scores: This position holds that high quant scores are a good indication of how crisply someone thinks, regardless of whether or not the discipline they are applying to demands any robust use of mathematics at all. The rebuttal maintains that unless someone is going to be working with numbers, the quantitative score can be completely discounted if the other two scores are high enough.
2. All or Nothing: Some reviewers of grad applications maintain that unless the GRE scores are quite high in all three domains, the student should be considered a bit of a risk. I've even been privy to a theory that links high-math/low-verbal scores to anti-social behavior. The high-verbal/low-math applicant sometimes gets dismissed by others as someone who talks a good game but doesn't have anything substantive to say. All style and no substance.
3. The Declining Significance of GREs: Some reviewers don't even look at GRE scores. They dismiss them out of hand. If the statement is strong and the letters are convincing/supportive, they don't need any other information. This position usually gets justified with recourse to discussions about the relative underperformance of minority candidates on standardized tests (whether that's chalked up to cultural biases written into those tests or to "stereotype threats" priming said students for failure). Of course, this anti-GRE position doesn't mesh well with university-wide attempts to clearly demonstrate the exceptionalism of incoming classes.
4. Writing, Writing, Writing: I've heard at least a couple of qualitative social scientists and humanists wax eloquent about the singular significance of the writing component of the test. (I don't even think we had a writing section when I took the test.) A high writing score means that whatever the prospective students know/learn will get translated into the coins of the realm in academia: the written conference talk, the term/final paper, the publishable article, and the Dissertation. This, they argue, is where the rubber hits the road for graduate studentry. And bad writers with good ideas have a more difficult time thriving in the academy. Good writers can survive even if their ideas aren't always instantiations of incomparable genius.
When I'm on a selection committee, I tend to go through all of the other materials in the files before I look at GREs. Starting with GRE scores can sometimes bias one's reading of the rest of a prospective student's application. Once I go through the written materials, I then compare my assessment of the student with his or her GRE scores, usually just to see how hard a case I'll have to make to my colleagues (if those scores are particularly low). Of course, I almost never win those low-GRE cases.