The point about supernatural phenomenons, weird experiences, that sort of thing, is that they are reported events - they have interacted with our world in some way - because someone, or several people have witnessed it - they have seen it, or smelt it, or felt it, or recalled it, or have had seemingly impossible facts about them related back to them by a stranger.
I would quibble a little with chakka, I think, who said anecdote was worthless - It has some indicative value, but as evidence it ranks at the lowest tier of what you might rationally require to conclude something is true.
So we can surely test for them - is the phenomenon repeatable? Does it affect more than one individual? How long does it last for? How frequently does it occur?
So when we talk about the failure of such phenomenon to manifest under scientifically controlled circumstances, or when spiritualists, astrologers, mystics, telepathists etc all fail to demonstrate any kind of truth or veracity above what might be expected for random chance, you really do have to question whether there is any kind of supernatural interaction with the real world at all.
So, whilst we might indeed be ignorant of some "energy level" or "other dimension" or the "afterlife", whilst we can think of no plausible way in which say telepathy might work, for instance, that does not invalidate the process by which you measure for such things.
We interact with the world via our senses; the brain makes up a lot of stuff. Our recall of actual events is incredibly poor - just read eye witness statements about the same event for instance, to see just how conflicted people can be. Our perception can be poor; Famous experiment told observers to watch a video of some basketball players in training, and count the number of passes and baskets carefully; they would be tested afterwards. The test question asked them was how many people saw the gorilla? Most said they had not seen one. But when the video was replayed, it was very evident that someone in a gorilla costume walked through the basketball players :)
With such easily influenced senses, with our poor recall of actual events, I do not find it surprising that people can become convinced they have seen or experienced something weird or supernatural....