Some of this has been said in some way or another, so ignore if you like.
My answer is no. Games cannot even hope to achieve the storytelling in the better books or films. But they shouldn't try to either.
Most of the, in my opinion, better stories are about characters, and not plot. There are quite a few games that have great plots, but a good plot does not a good story make.
Where books and movies tend to differ is setting and introspection. Books usually have a lot of time to expand on these, making the reader very aware of why the characters do what they do in the story. Movies usually have a lot less time for this, and cannot use some techniques that books can to convey thoughts or emotions. Instead they have to rely more on pace and direct visuals. I think that this is why you hear 'the book was better' so often. Movies, on the other hand, provide for a much faster and easier 'entertainment-satisfaction', and they can convey pages of a novel in just one 3 second shot, if done right.
But just as some conventions used in books would be really awkward on the big screen, some story-telling devices come over as weird or annoying in games. There are games that try to tell a good story (and perhaps do well), but as a player, you may be wondering why you haven't needed to press a button for half an hour, or why you couldn't steer the conversation to a different direction than 'yes, no, i kill you'.
Rather, the better 'story' games i know of are about immersion. They present you with a world with which you can interact. And it is this aspect that i think they should be focused on. You cannot see what happens when the main character of a movie does something else than what he does in that movie. You can in games. Now unless they can come up with a full world simulator, games will always be held back by what the developers can come up with, but they still allow for a lot of experimentation. Another great aspect is interaction with other players, which is also not possible in movies and books. Be it to shoot 'em in the head in a deatchmatch or team up to go beat that big dragon.
I think that this is also why some movie/book to game conversions do work; you know the world, go play in it, while most game to movie conversion do not. They could, though, if they would for instance take that world from the game, and think up a separate story inside it.
Each medium has its pros and cons, each medium is great, and they should all do what they do best.