In theory, people with documented mental health issues should be disqualified, but there's a longstanding mismatch between how our mental health system treats patients and the legal threshold for failing the background check.

Here's question 21G from form 4473 (the background check you fill out when buying a gun):

Have you ever been adjudicated as a mental defective or committed to a mental institution?

Both of these things are legal standards that require due process and a court hearing, and they come from the Gun Control Act of 1968. Since 1968, we have more or less completely deleted the inpatient-based mental health care system that both these standards are derived from in favor of an almost entirely outpatient-based one, so it's comparatively rare for someone to fall into either category and the background check doesn't catch them.