I like to think of myself as a non-dogmatic intellectual with an open mind. I tend to interpret stories about mystical or supernatural experience as phenomena of the mind that may have some metaphorical truths contained within them, while not actually being literally "true". Dreams are another example--often times dreams can be intensely meaningful and relevant, yet my standard belief is that they are not literally predictive of the future, etc. I like to think of such experiences as most often being neutral structures of the mind, onto which we can consciously apply or project meaning, as opposed to being intrinsic sources of meaning. Dreams are like the canvas; our conscious mind may apply direct meaning to the canvas.
There are some specific instances, though, in which some of these phenomena can influence subsequent events. For example, if thousands of people read the same horoscopes in a newspaper, and the horoscope advises Capricorns to wear red, and to be prepared to smile and make friends with a stranger wearing yellow; and Libras are advised to wear yellow, and to be prepared to make friends with a stranger wearing red -- then you can see that the horoscope may end up influencing subsequent social behaviour, and the prediction may come to pass with greater likelihood. If it does come to pass, the Capricorns and Libras who meet each other will even more strongly believe in the mystical power of the horoscope. Perhaps some other mystical experiences can work in a similar way, that the heightened expectation leads to a higher likelihood of certain experiences coming to pass.
Yet, I have to acknowledge, in my work with many patients, that there are experiences in the mind in a sort of mystical realm, that can be very significant, eerily predictive, powerfully moving, and unexplainable through conventional logic. There are some people who seem to have these kinds of experiences more often than others. A quality of these kinds of experiences is that they may defy logical testing, replication, or "proof". I think we should all be open-minded about this, and embrace the truth that we can't explain everything, perhaps some things are intrinsically unexplainable, and while we should never suspend our intellectual faculties, we may sometimes have the opportunity to embrace, enjoy, or be moved by, a special "mystical" experience. History is certainly full of stories of individuals moved and inspired by such experiences.
Due to the elusive nature of such experience, though, I do think it is important not to fall into dogmatic interpretations if they should happen. Delusional or psychotic phenomena are quite frequent, and when such experiences strongly influence behaviour, or are held to be literal external truths, there can be all sorts of problems. I think what distinguishes an unexplainable or poignant "mystical" moment from a delusional belief is that the former is poetic or symbolic, and the meaning is understandable by peers, while delusions lead to concrete, ongoing changes of belief and behaviour which are highly dissonant from the perspective of others (e.g. a delusion that aliens are monitoring one's movements with cameras and secret agents).