The problem comes when enough of the server is up to do a health check, but not enough to actually serve the request. For instance, a Java app server which has one app to respond to health checks, and another app to do processing - the second one can be jammed while the first one's happily reporting that everything's okay.

If you then move the health check into the main app, maybe it eliminates a lot of false positives but the same thing can then happen further into the stack...