Let's take the example of the exorcisms of Jesus. Many modern people see psychological and neurological illnesses as the causes rather than demons. Jesus apparently believed he was casting out demons and even thought he was talking to demons sometimes. God was healing people through Jesus even though Jesus didn't understand what was actually happening as thoroughly as we might understand today.
I don't see a problem with believing Jesus was unschooled, ethnocentric, superstitious, or anything else that would have been typical for a Jew of his era. Jesus through obedience to God might have overcome his human limitations through perfect obedience to God, so that God could say "this is my own son in whom I am well pleased, listen to him."
This keeps sounding like Adoptionism to me, which I certainly disagree with, as I consider Jesus the Incarnation of God Almighty.
Anyway, we are running into problems of interpretation here. You assume Jesus was curing neurologic or psychologic disease, because you are reading that into an explicit text of Exorcism. We have no way of determining the fact of the matter behind the text, whether this was a cure of such disease or not. It certainly appeared to the writer, or the source he derived from, that Jesus was casting out demons. This doesn't really tell us what Jesus Himself thought He was doing, so in essence the argument is trying to read between the lines and behind the text.
Regardless, it is not uncommon for the mentally ill to consider themselves possessed, sometimes even amongst the secular (hearing voices, command hallucinations, etc. are common symptoms). If Jesus 'cured' the mentally ill by commanding their own delusion to depart, there really is no problem deriving the text from such an act (barring the demoniac of Gadara's pigs).
This all assumes that there aren't demons though. If demonic forces really were out and about, I would think the mentally ill to be low-hanging fruit, ripe for possession. Their faculties are already impaired, so why not? The difference between the Possessed and the Mentally Ill would thus be frequently blurred. So unless we start with the a priori assumption of Materialism, when examining a text that clearly makes Spiritual claims, you would not think thus. This seems a tad silly to me.
Further, the argument is often made that we "don't see possession today", but if we are dealing with beings with volition here, perhaps they are catering to the audience? It would fit the aims of presumed demons to run under the radar today, as that great quote of the greatest trick of the devil being us not believing in him, has it. In a time where everyone did, it would make more sense to try and demonstrate their power. A great literary treatment of this, is the devil in Brothers Karamazov by Doestoyevsky - where Ivan doesn't know if he is real or a delusion of his fevered mind.
So, really it depends how the text is interpreted, so is dependant on the readers' assumption and worldview - rather than Jesus'. Only the Pig episode, as far as I am aware, unequivocally endorses the existence of the demoniac therefore. If you are so keen to read modern Psychiatry and Psychology into the text though, it would be fairly easy to dismiss this as embellishment. In my personal opinion: If demons exist, there is no problem with the Biblical text; if they do not, then mental illness in that cultural mileau would present as demonic possession as depicted regardless - so either way, Jesus' action seems explicable, the differentiation blurred at that stage, so the answer to which Jesus 'really believed' says more about the metaphysical or historical assumptions of the answerer, than the opinion of the man Jesus. We all concur the writers of the Gospels believed in demons, but that doesn't mean Jesus did, unless you imbue Him with your own presupposition as to His nature.
