Faith does work sometimes, but it's explained easily. Two words: placebo effect.
Placebos are used routinely by pharmaceutical companies to test the effectiveness of their drugs. It is a widely known fact that the brain can have an influence on the body. Expose a person to something they truly believe is poison ivy, and they can have an outbreak. Belief at work.
Concurrently, if they know it's not really poison ivy, they will not have an outbreak. This would be a non-event "caused" by the lack of belief or faith.
Psychosomatic illnesses can be healed if one has enough faith in the element of healing. Nobody will ever be able to regrow a leg, however, based on faith alone, because it is not a psychosomatic malady.