There seem to be a number of people who believe all things are caused by demons or angels (not calling out anyone specifically just someone's comment made me think of this again) so like, would someone with Schizophrenia or Dissociative Identity Disorder ... have demons? Need an exorcism not pills? This is very frightening to imagine
We were born from a cused couple - Adam and Eve were corrupt before they had children. Therefore, we are corrupt.
Corruption does not become polishment: that is a violation of one of the most important laws of physics
as we known them. As such, we will always continue toward corruption.
Christ's sacrifice not only stops the corruption, but it reverses it for those who believe and follow Him/God the Father!
The Holy Spirit "marks" you spiritually, and with God, the Christian does not get possessed. Remember, the spirit is willing, but the flesh is weak. Ailments of the flesh (including mental issues - which is still flesh,) are a consequence of our corrupt nature.
Christians can, however, experience severe oppression by spirits. For example, spirits can inflate their own "power" by causing anxiety about the possibility of physical ailments being demonic in nature. There was a time when the greater Western scientific world thought physical illness was caused by demons - which we now call "germs (microbes)."
Spirits often oppress Christians through physical imperfections that already exist with them in the first place. That is part of
spiritual warfare. It is often simply a "spiritual battle" mistaken for demonic possession. To effectively fight, you need your weapons and armor from Ephesians 6:10-18.
Having said all of that, part of spiritual warfare is also being "as wise as a serpent...," because demonic spirits are obviously intelligent - much more than us. In fact, it would be a mistake to underestimate them. Many possessed people do not realize they are possessed, or what are called "Walk-ins."
But, someone who has the holy Spirit, and is deserving of it (i.e. has accepted Christ, believes, etc.) - I would dare to say cannot be possessed. Why not? Because the void has been filled, and no evil spirit can "boot" the holy Spirit out of you once it has set up house.
Mental illness should be treated like any other illness. Nutrition is first, then well-being, medical... God first before all of it. Moreover, in the same way you would be vigilant about medication for any other thing, don't let the stigma of the illness being "mental" make you hasty in a "fix."
Often times our "illness" as seen by society especially, are actually gifts. But, if it is interfering severely with your life and spirit(uality,) then I would say start with God first, and really hear Him. Don't be hasty, or scared of what answer you get.