GameSpot confirmed with Double Fine that it has nothing to do with the Double Fine Adventure:
"While we can't comment on unannounced projects, we can say that Double Fine's Kickstarter-funded adventure game has not been started, designed, or even imagined yet. No trademarks have been filed on its behalf."
There's been some suggestions that it might be the game Ron Gilbert is working on with Double Fine, but really, it's anyone's guess what The Cave really is. Hopefully it has a better fate than LucasArts' Handsome Halibut.