I'm pretty sure I've read a story about a psychiatrist who found out that some government agency (not sure if it was NSA) was illicitly browsing his patients' records and particularly their prescription histories without any kind of consent (as would be the case for a security-clearance audit). Their excuse was that some conditions might affect people's ability to perform in security-sensitive roles. It's also not unlikely that they might use the threat of disclosing someone's medical/psychological status to coerce keys out of them. So yeah, NSA might well want that information enough to "suggest" adding a back door.
The SF86[0] (the form you'll be filling out if applying for any type of clearance) is extremely "intrusive" with regard to your medical history. Excluding information isn't an option, either.
