Consider EA Sports, BF2, Counterstrike, or Civilization. They are all basically sandboxes that allow players to have an experience. The exact experience changes with each play, and the per-game story can be suspenseful and have twists. Can this type qualify as art? I don't think so. They are toys to play with.
It's not so cut and dried though. They can have artistic elements in them. Most games have "art assets" and artistic direction with sound and music to provide an atmosphere. It's hard to believe there can't be ANY artistic value to any of that.
But then there's the plot games. Interactive fiction (all the point and click adventures are basically just graphical versions of this). They present a story; isn't it conceivable that that story could be art? Even if it almost never is. The main issue is that the player interaction is often "pointless" (Dragon Slayer arcade game: the story plays but the player has to exercise reflexes to keep advancing the story).
So, assuming LOTR is art, would it still be art if you have to get through a hand-eye-coordination exercise or solve a Myst-like puzzle to advance the story? It's not clear how interactive sequences affect the art.
I think potentially, interactivity can draw the viewer into feeling like he is "in" an experience, rather than watching one. Half-Life 2 would be relatively boring to watch as a video of someone playing through. But clearly there is an experience playing through it... you are directly controlling the character and feel like you're in another world.
But computers are so limited in what you can directly control. Action is the order of the day because we don't have AIs with language and speech processing. Furthermore even if we had that, what would drive the player to do interesting scenes? Look at the movie Taxi Driver. You can't make an interactive experience out of something that is based on an intricate character. Game personas are either shown in cutscenes or provided to you in conversation choice items (or you sort of indirectly control a strong character, as with Full Throttle).
TLDR: I think most games that are art, aren't art because they are games. The interactivity that directly contributes to artistic games is nice but ultimately limiting in the nature of experience they can provide.