When Should I Take the GRE for the Fulbright Scholarship?
The short answer is anytime before the Fulbright deadline i.e. May 11th 2016. It is not mandatory to send or attach the official GRE scores before the application deadline. You can even take the GRE test one day before the Fulbright deadline and send your unofficial scores to the Fulbright program. Unofficial scores can be entered in the test scores section of the application form.
Do note that once you take the GRE, you have to send the scores from the ETS directly to the USEFP and you can do that by entering the code 9388 while sending your official scores to USEFP. If you plan to do this on your GRE test day, you can do that for free after you have taken the exam and seen your scores. If you plan to do it after your GRE test day, then you will have to enter the same code but pay an additional 27 USD to send the official score card to USEFP.
In case you cannot get a test date before the Fulbright deadline, then you can also attach the GRE registration slip (the one that you would get through email after you register for the GRE) with your application form and send it to USEFP. Fulbright does allow candidates to take the GRE after the Fulbright deadline only if test dates before the deadline are not available.
Your application to Fulbright will be discarded if you do not take the GRE before the Fulbright deadline or if you do not register for the test before the Fulbright deadline (this is only in case you cannot find seats).
Some students take their GRE exam well before they submit there application. In that case, they can send their GRE scores before sending their applications to USEFP. USEFP will retain your GRE scores with them even if your application is not received.