Legality of a mentally disabled person owning a firearm

The Brady Handgun Violence Prevention Act states that it is legal for a person to own a firearm who has been as long as they–these are the exact words—“have not been adjudicated as a mental defective or committed to a mental institution”.

Does that mean that is illegal for someone who determined to be disabled by Social Security because of mental illness to own a firearm? Does that qualify as being “adjudicated as a mental defective”?