The inability to prove God doesn't exist is a universal epistemological limitation and has nothing to do with the likelihood that God exists.
You know--I hear this a lot. And in my opinion, that's just not the case.
A lot of people seem to have convinced themselves that "you can't prove a negative," or "you can't prove that something doesn't exist."
Neither of these claims is true. You most certainly can prove a negative. In fact, proving negatives is fun and easy.
The simplest way to prove a negative is by denying the consequent:
If P, then Q.
Not Q.
Therefore, not P.
If there was an elephant in this room, I would see it.
I don't see it.
Therefore, there is no elephant in this room.
This type of argument is called a modus tollens. And two of the most convincing arguments against the existence of God--the problem of evil, and the problem of disbelief--can be boiled down to simple syllogisms like these. Namely:
If God existed, then everyone would be good. (Or: if God, then universal goodness)
But not everyone is good. (Not universal goodness)
Therefore, God does not exist. (Therefore, not God)
And:
If God existed, then everyone would believe in Him. (Or: if God, then universal belief)
But not everyone believes in Him. (Not universal belief)
Therefore, God does not exist. (Therefore, not God)
These are both valid arguments. If both of their premises are true, then their shared conclusion must follow. And since, in both cases, the second premise is obviously true, they can only be unsound if the first premise is false.
Personally, I'm convinced that, in both cases, the first premise is true, and that the God described in the Bible does not exist--indeed, cannot exist. I've read and heard numerous counterarguments from religious apologists, and I haven't found any of them persuasive.
I don't know what other people are experiencing when they claim to experience "God." I don't think anyone knows for sure. But I'm pretty sure I know what it's not.