One could argue that despite Limbo not having a narrative it could tell stories just by locations you visit. Stories don't always need to be told through dialogue, and there's one game I'm thinking of that does it but I can't even remember anything about it at the moment, and it's not really the focus for this topic so whatever =p (Coming back to this, it was Journey... though that didn't have narrative or dialogue so to speak, but it did sort of show the story through hieroglyphic image sort of deals. Does that count? whatever) (EDIT: Also, Flower... made by the same guys I think... HMMMM...)
I personally figured that 'adventure' games had an exploration element to them...
"Adventure' isn't a genre I see by itself really, at least it's not listed like that. Usually it's "Action/adventure" or "Point and click adventure" etc. So, what makes an action game different from an action/adventure? I do see 'adventure' genre listed in places like Steam and GoG but there are many varying types of games in there I don't know if it really counts. I see 'Antichamber' 'Assassin's Creed' and 'Far Cry 3' in that category, and I'd consider those games to... not really be the same... kind of game.
If you want to be fancy and technical, here's the definition for 'adventure'
a : an undertaking usually involving danger and unknown risks
b : the encountering of risks <the spirit of adventure>
: an exciting or remarkable experience <an adventure in exotic dining>
So how does that apply to games? You could argue that many games involve an adventure of some sort if you're going by the definitions above, whether it be a platformer or a first person shooter. In fact, I don't recall Limbo ever being considered an 'adventure' game, but rather a 'puzzle platformer,' a genre that has become fairly popular with 'indie developers' it seems. Despite this, you do go on 'an adventure' wherein you 'encounter risks' during an undertaking (platforming). I can't speak much for The Cave as I've not played it, only seen someone else play it for a bit, but you do encounter risks and you do go on a journey of sorts.
Far Cry 3, Assassin's Creed, and Antichamber all have exploration elements to them, and you go on adventures of sorts...
It seems maybe 'adventure' as a genre is just too vague? They probably could have just had a genre called "Game with protagonist(s)"' and you'd get many of the same games in there as well it seems.
Bleh, topics like this are kinda difficult =p