Yes all believers and non-believers have experiences. When a believer experiences something that is taught in Scripture - such as the New Birth (John 3:1-10), then this experience is constituted as experiential knowledge. To put it another way: It is one thing to have head knowledge of a subject, quite another to have experienced it.
However, experiences never, never, never prove Scripture!!!!!!!! ALL experiences are weighed and judged by Scripture as to whether are not they are from God.
From the perspective of the unbeliever who comes to faith, an encounter like this might be perceived as a kind of proof, or evidence at least. There are many examples in scripture that show people believing or at least paying more attention to the message after seeing miracles.
Here are a couple of verses from an account of a healing in John 4:
48 Then said Jesus unto him, Except ye see signs and wonders, ye will not believe.
...
53 So the father knew that it was at the same hour, in the which Jesus said unto him, Thy son liveth: and himself believed, and his whole house.
Sergius Paulus believed after seeing Elymas struck blind. God bore witness to them that heard Christ through signs, wonders, and gifts of the Holy Ghost, according to His own will.
If a person claims to be able to speak in a language not formerly known by the speaker, the language must still follow legitimate linguistic patterns. The ecstatic utterances I have witnessed are pure gibberish. What they speak does not harmonize with Scripture and therefore can not have it's origins in the Holy Spirit. They either have their roots in the energy of the flesh or in the energy of the Devil.
My bachelors degree is in Linguistics, and I've studied a number of languages myself. One question that comes to mind is how many people who say such things are really qualified to determine if a language is pure gibberish? Cantonese, Vietnamese and especially Hottentot sound like pure gibberish. But English, Indonesian, Malaysian, and Spanish sound like real languages. I know or have studied a bit of the latter four, but not the first three. What is gibberish is a matter of perspective. If you do not know a language, it may well sound like gibberish. One critic of speaking in tongues on this forum some years ago wrote 'harga warga' mocking speaking in tongues, presenting that as an example of gibberish. I informed her that was a couple of words in Indonesia-- 'price' and 'family.' She couldn't tell even the gibberish she made up were real words in some language, so how could you say that something is gibberish and not a real language?
It may be that some analysis of a given case of speaking in tongues does not fit with a particular linguist's understanding of how human languages are inflected for meaning. Paul also suggests the possibility of tongues of angels. He doesn't say he did it. He suggested it as a hypothetical possibility. We have no idea how angelic languages are inflected for meaning. My understanding is that the linguist Samarin's work argued that the intonation of speaking in tongues did not fit the pattern of human language, but I have heard people praying in English in the same high pitched monotone, and if I were to follow his reasoning, I'd have to dismiss that as non-language, even though I understand it and it is my own language. He must have had a peculiar sample of recordings, also, because the high pitched monotone is not typical of speaking in tongues in my experience.
I would not say that all that claims to be speaking in tongues necessarily is. It could be that there is some human-generated glossalalia that is not the real deal. But I do not see any warning in scripture that speaking in tongues will be demonic. I do see Christ's teaching against the idea that if one asks the Father for a fish, that the Father would give Him a serpent. Also, the idea of praying to receive a gift of the Spirit seems quite in line with the teaching of scripture, since I Corinthians 14 encourages believers to desire to prophesy and directly encourages those who speak in tongues to pray that they may interpret. In Acts 4, the apostles prayed to God to do signs and wonders and to stretch forth His hand to heal for the sake of His Servant Jesus.