This is perhaps the most coveted question when it comes to the Fulbright. Any score above 310 is a ‘slightly above average’ score, a score above 317 is ‘an above average score’ and any score above 321 is ‘a really good score’.
Speaking from experience, I know students who have secured the Fulbright despite having scores ranging from 295-309 in their GRE. What this shows is that even if your GRE score is low, you can make up for it with a high CGPA, solid letters of recommendations or stellar Personal Statements and Statements of Objectives. Despite what most Pakistanis believe, the GRE does not make-or-break your Fulbright scholarship. Also, if your GRE score is too low, you can still be selected by the USEFP and be asked to retake your GRE after selection. This second GRE is funded by the USEFP, so you do not have to worry about costs.
Just remember to not spend a lot of time going over Kaplan, Collegeboard or Barrons as these are redundant sources that do not reflect the actual difficulty level of the GRE. Moreover, if you are applying for an engineering or science degree, your quantitative score will matter more than your qualitative one. The opposite is true if you are applying for a degree in the humanities or social sciences.