Is the game 2048 always solveable?

On the contrary, I think the game is not solvable. You can get convinced of it by taking a look here: http://sztupy.github.io/2048-Hard/ It is the same game, except that randomness is replaced by the worst possible choice for new tiles to appear. It seems obvious after a little practice that this version is impossible, and it just corresponds to being unlucky in the original game. A rigorous proof that this version is impossible will likely be very tedious though...

It was claimed that the 2048-game is NP-hard in this arXiv preprint: http://arxiv.org/pdf/1501.03837v1.pdf