Many people swear that sometimes, especially when they are either falling asleep or just awakening, they have strongly felt that somebody or something is in the room with them, even though they should be alone (or even if they are with somebody, they feel that a third person or something is there.) This often startles them, and when they awaken, it's gone.
Religious and superstitious people attribute this to spirits, ghosts, demons or other supernatural beings. More rational people attribute them to what they really are: Hallucinations.
Hallucinations are much more common than people realize. They usually happen precisely when a person is just falling asleep or about to awaken. They are very closely related to dreaming, and could be quite sensibly be classified as a type of dream. They are normal and they are in no way indication of mental illness or anything of the sort. Most people have experienced them at least a few times during their lifetimes.
Another form, a rather different one, of hallucination is a strong sense of a great epiphany. The sense that something really deep and fundamental suddenly makes complete sense to you, that you suddenly have a clear understanding of something that's extremely mysterious, unknown, or grandiose. This feeling is often described as "being one with the universe" and other similar things. Sometimes it can be related to something much more mundane or personal, perhaps something that one has been thinking about or something about oneself.
Almost invariably, when the person then awakes fully, they can't remember what that great epiphany was. They just remember the feeling, but not the details. They feel that they had a great answer to something, an amazing understanding of it... and then they just forgot what it was.
Again, this is just a form of hallucination. It's not really that the person understood something or got the epiphany; it's simply that they got the strong feeling that they got it, without actually getting anything. It's not like if they could just at that second write it down or tell somebody, they would have gotten something coherent because there really wasn't anything to say. In actuality it was nothing, just a feeling. In a sense, it could be classified as a momentary delusion caused by the hallucination.
These are some of the sources of such claims, and one of the reasons why some people believe in the supernatural, in having some kind of supernatural senses or powers, or being "connected" to something superior or supernatural.
In reality it's just their own brains playing tricks on them, and them interpreting it as something it wasn't. (As said, this is not indication of mental illness or anything of the sorts. It just happens and it's normal. One should simply realize what they are and take them as such.)