This is actually a pretty solid theory. If
the Ring is to be used to get someone what they wanted, whether just invisibility like in the cases of
Smeagol and
Bilbo, to gold (in the case of
dwarves), to power and strength (in the case of
men and perhaps
wizards), then, to somebody who was totally content the way they were and without a care in the world, the Ring would have NO effect whatsoever on them.
Granted, I think the first time Bilbo put the Ring on, he wasn't intending to make himself invisible, so it's not a solid argument that if you don't intend to use the Ring to do something, that it won't still do something to you. However, to be fair, on that one, now that I think of it, Bilbo WAS intending to NOT be found by
Gollum, and the Ring obliged. As for
Isildur, from what I can tell when he wore the Ring, he was intending NOT to be found by the
Orcs (and the Ring obliged for a while, then betrayed him.) Had Bilbo found the Ring and not intended to hide from anyway or be invisible, perhaps it wouldn't have made him invisible.
Of course, on the other hand, I don't think
Frodo was intending to be invisible at
The Prancing Pony, but perhaps he'd had too much to drink and in his drunken mode, actually DID intend to disappear as a childish prank.
Anyway, as
Bombadil doesn't seem too bothered that
Sauron is on the loose or that he has
evil Hourons on one side of his house and
Barrow Wights on the other, I'd say he's totally happy the way he is.
As such, perhaps for that reason, the Ring is useless against him. Ironically, for that reason, it would indeed make him a very poor guardian of it. Either he would get bored of it and toss it away or caring for it enough to destroy it would get him to actually desire something he doesn't have (the destruction of the Ring). Hence, the Ring would try to tempt him to use it to make it easier to get to
Mordor so that he could destroy it (which, would, of course, in time, end up backfiring by revealing him to Sauron, which would have been what the Ring really wanted.). Also, for that matter, it's also likely why
Galadriel and
Glorfindel didn't take the Ring. Even if they had the sense to know that keeping it as a
weapon to defeat Sauron, rather than destroying it, was a foolish idea, the Ring could have gotten to them to use their powers to blast through foes in their way to Mordor in their quest to destroy the Ring, which would alert Sauron, who might even end up coming himself to waylay them.
Anyway, because Bombadil doesn't desire anything from the Ring and is content the way he is, the Ring doesn't affect him. As for how he was able to vanish it, I have no idea.
BTW, I think even
Gandalf confirms my theory when he says "say not that he has power over the Ring, rather, that the Ring has no power over him."