Is it OK to ask the graduate chair if my background is competitive? I'm planning to apply for a M.S. in some Universities at the US, and at one of those, I tried 2 times and failed (bad GRE scores). Now I reapplied, and got better results (Q:159, V:154, A:3.5), but not sure if it will put be in better shape than before. One thing I think it can be a differential is the fact I had an exchange program back at that university. Is it ok to send my grades to the graduate chair and ask if my application is competitive enough?
I am currently the Graduate Coordinator of my department (mathematics, UGA), and it is the season for me to get a lot of emails from would-be or might-be graduate students.  Variations on "Will my application be competitive if I apply?" are the most common question I'm getting.

It's okay to ask this question in the sense that it doesn't bother me at all and I will reply with something.  On the other hand, my reply in most cases has been along the lines "Thank you for your interest in our program.  I encourage you to apply."  I don't think it's in anyone's best interests -- mine, my department's, or the questioner's -- for me to try to evaluate the degree of competitiveness of an application based on very incomplete, self-described information.  I can (and do) address the required components of the application.  If based on the information provided it seems like applying to a graduate (usually PhD) program at a top 50 math department may be a mismatch -- e.g. the applicant's undergraduate major is in something other than mathematics (or a closely related field with many mathematics classes taken) -- then I might explore that a bit in the reply.  But the bottom line is that the application process is relative and competitive, so more than likely we can't know how any given applicant will fare until we see all the applications: i.e., until we go through the application process.  (However, if the self-described profile of the applicant sounds very strong, then I may mention that and encourage them more enthusiastically.)

A few final comments:

1) I can see that it is frustrating to apply, perhaps repeatedly, without knowing one's chances.  But in fact the state of applying for a bunch of things without knowing whether one will get them is a quite common one in later academia, so maybe it would be well to wrap your mind around it now.  (It is not a trivial thing to wrap one's mind around.  I was a really excellent high school student.  When I took the PSATs I expected to become a National Merit Scholar, and I did.  I applied to the University of Chicago not just because I expected to get admitted but because I expected to get a merit-based scholarship, and I did.  I had enough of this kind of expected success that when it came to optional academic competitions / scholarships for which the outcome was in real doubt -- and especially when I didn't see success improving my life in any clear way -- I often didn't apply / show up / give my best effort, to the point of occasionally disappointing my mentors.  Looking back at all this, I must say that I find my behavior quite rational.  But it's not the way that "grownup academia" works.) I spend weeks every year writing a big national grant application; thus far, I've applied for the grant five times, gotten it once, not gotten it three times, and the latest time is pending and will be for many months.  For this grant (NSF), I do get feedback every year, which is helpful overall (though sometimes frustrating in its particulars).  On the other hand, I have applied twice for a smaller, privately funded grant (Simons).  When I didn't get it the first time and got no feedback about it, I was pretty taken aback and felt like complaining about it.  But I thought it through: it was clearly to my advantage to suck it up and apply again.  I did, and I got the grant the second time.  That's grownup academic life -- relatively successful grownup academic life, in fact.

2) There is a kernel of advice lurking in point 1) above.  The best time to ask for feedback is soon after someone has carefully evaluated your application.  If you've applied to a program and didn't get in, you have every right to ask why and how you could improve your application.  It sounds like you think that the reason you didn't get in were your GRE scores: is that what you were told?  If so, given that you improved on the thing they told you to improve upon: that sounds good, you should probably reapply.  Finally, if you already spoke to someone at the program, they told you to improve your GRE scores, and now you have, then it makes considerably more sense to me to continue that conversation with the person who told you to improve your scores: given that your GRE scores have improved so-and-so much, do they think you should reapply?  In that situation you have a better chance of being told something substantive, I think.

3) My program is relatively small (we generally enroll no more than 15 students per year).  In quality, the distance from the very top programs and from "less than serious" PhD programs are each rather large.  Once in a while we "steal" a student from a top twenty program, and sometimes for various reasons we take chances on students with inferior backgrounds.  So our admissions process really plays out a bit differently every year.  I could imagine that a program which is significantly larger, better or worse than ours might have less variation.  I suppose I would nevertheless be surprised if such departments were significantly more forthcoming in their reactions to questions like this, but I don't really know.