Agreed with everyone else. They don't have access to your records so you totes could fudge, though i really doubt they'd care if you went to some couseling. there's the part where the doctor checks off that he/she thinks you're fit to conduct research abroad, so they could figure out whether you're totally insane there.
Speaking of health stuff, insurance. Is the Fulbright-associated insurance suitable to be one's only source of insurance, or is it more intended to be a supplement? Maybe I'm dumb, but I feel like i'm getting mixed messages from the fulbright website.