The real problems arise when we are trapped into circular patterns by our theology. When we are taught to believe that our problems are all caused by demons (when they aren't), and that it is our lack of faith or holiness that keeps us oppressed. Or when we are sick and God is using the situation to refine us, but we cannot accept that because we think that He wishes everybody to be healed all of the time. So we turn our attention inward, and consider ourselves to be failures and unworthy of the graces of God.
Generally a person who prays for the sick/oppressed and doesn't see them instantly healed, will look for other means to resolve the problem. People for centuries have found the need to walk the Christian life, and thereby see the deliverance from the bondages of the World. Resist the Devil, and he will flee from you. Sometimes this takes longer than others. And if God doesn't choose to miraculously heal you, perhaps His plan is to lead you to the right doctors, through whom He plans on solving the problem? Or letting you be a light to others who also suffer in the same way, knowing that this life is but a flash, and that spiritual health and wellbeing is a much greater prize than physical wholeness???
The opposite is true though as well. I have seen more traditional Christians struggle with sin for years and years, never considering the possibility that there may be demons involved that could be rebuked and quickly cast aside. I imagine the same goes for Healing. Sometimes we struggle needlessly, when all we have to do is come to God and ask for wisdom.
That right there is the key. Seek, and you shall find. If we come to Him, He will show us which of life's trials have an easy fix, and which ones we must learn how to die to ourselves through.