Just chiming in to provide an outlier story. If applications with low scores are thrown out in the beginning I would never made it to the finish line, which I did. I took the (old) GRE four times in four years, and my quantitative score never got above 420, and my writing never above a 3.5. On one remarkable day I got a 790 on verbal, but usually my verbal score hovered around 500. What the scores as a set show is that I do not excel in a test-taking situation. No more, no less! I am actually quite good at abstract mathematical logic problems, but you'd never know it. My essays and letters were outstanding, and my GPA was ok but not great (around 3.0). Over these years I was getting rejection after rejection and grew convinced I had to raise my GRE closer to the 600/600 mark to round out an uneven academic history. The last year I took the GRE I completed two formal classes (PM me if you are interested), studied with an informal study group on weekends, and hired a UC Berkeley undergrad to tutor me in math. For a few months I made this my top priority. On the practice tests I was consistently scoring close to 700 on both sections, and thought I had it in the bag! But in a cruel twist, I got exactly, to the number, the same score as my first test four years before.
The bright note is that I got in to a very good PhD program last year with six years of full funding (which I turned down for personal reasons), and two masters programs, NYU and Chicago. Berkeley told me I made it to a shortlist back when I first applied in 2010, and mentioned I should work on my GRE score. But.... I made it to a shortlist!!! My GRE scores have not come up when I've asked for feedback after other rejections. In short, I wish I hadn't agonized about it so much.