A game can be "artistic" in different ways. And frequently, games are artistic in more than one way. It can have an artistic touch to its visuals (eg.: Valkyria Chronicles, Okami, Flower); in that aspect, those games are comparable to the plastic arts. They can be like "paintings in motion". If you analyze them as such, you can't say they're not art, because their aesthetic is as artistic as most paintings, which are considered art.
The game can feature complex and deep Storytelling. The storytelling can be interactive or not... a positive point of videogames as storytelling methods is the ease with which they can make the storytelling experience interactive. On the other hand, being "interactive" is just a plus... The important thing is that the storytelling has to be WELL WRITTEN. Player Interaction in the storytelling is just a gimmick, if the core content of the storyline isn't developed enough. A good example of this are the "game books", interactive novels in which the reader can make some choices and follow some instructions in order to have an interactive reading experience ("if you want to say yes, go to page 28", "if you want to say no, go to page 122" kind of thing). Those "game books" aren't given a lot of importance on the literary medium, as far as literary value goes, because the authors are always so concerned with the interaction, that they forget to create a good story. The day someone manages to create an interactive story in which all the possible "read-throughs" (my term for "playthrough" in an interactive book) result in a storyline worth of a nobel prize, then maybe the "interactive art" genre will receive more respect.
Back to the "games with good storyline" subject... those can be comparable to movies and books. A good storyline is made of good dialogs, memorable scenes, charismatic characters and good plot. There are a lot of examples in this area... games of all genres in which the storytelling is so well-written, it is comparable to that of award winning movies/books (Eg.: Metal Gear Solid, Siren, Deus Ex, Ico, Shadow of the Colossus, Valkyria Chronicles, Shenmue, Final Fantasy Tactics, Snatcher, Heavy Rain, The Darkness, Chrono Trigger, Persona 2, Silent Hill etc).
Last by not least, the game can be artistic because of its gameplay. Sometimes, this feature is linked to the game's visuals (like in Flower, for example), when the innovative gameplay is contributes to the artistic value of the visuals. Sometimes the storytelling can benefit from the gameplay, when said gameplay helps the storytelling to gain an additional surge of emotion, or even adds some complexity to the storytelling method. A good example is the microwave scene in Metal Gear Solid 4; the complex Link Navigator and Archive system present in Siren, which you have to figure out, like a puzzle, in order to understand the events chronologically; the interaction present in Heavy Rain, that doesn't make the storytelling generic, and really adds a completely different storytelling for every choice, rather than giving you the false impression that you can affect the story, but very superficially (eg.: Fallout 3).
The final type of "arty" game I can think about is the game that is artistic mainly because of its gameplay. I talked about games in which the visuals and storytelling benefit from the gameplay and thus become even more artistic. But... the gameplay CAN be the main artistic aspect of the game. This is the most controverse aspect that can define a game as art. With given arguments, the other types of games can't be discarded as art, because they're comparable to movies, books or paintings. The part of them there is art, is the part that is like movies, books or paintings. But to consider the gameplay aspect as the core artistic aspect of a game is very risky, because you have no paramaters, nothing artistic to compare this game to. It's like a TOTALLY new kind of art... and many disagree that an innovative gameplay experience is something worthy the title of "art". As long as it's something visual, it just has to be as good as an artistic painting. If it's the story, is just has to be as good as the story of an artistic book or movie. But what about gameplay? What would make gameplay artistic? What are the parameters? This is an area of debate. Examples of games that try to be artistic by their gameplays are Braid, Portal, Psychonauts, Katamari, Noby Noby Boy etc. Most of them have some artistic "visual" content, and even "artistic storytelling" to some point, but one can't deny their focus is on innovative gameplay, and they rely on that to call themselves art. It's not something unanimous that they are art. I, for one, believe they are.
So, if I had to choose a paradigm of "art game", I'd either choose Metal Gear Solid or Valkyria Chronicles, for great storytelling excelence. Their storylines are better as storylines then most "visual appealing" games are good as paintings, for example.
Edit: Forgot to talk about games that are "musical artistic experiences", comparable to the "music" art. There aren't many. Not any musical game can be considered an artistic experience. Games like Guitar Hero/Rock Band/Dj Hero, for example, IMO, don't deserve the title of art, as they only have songs in them, but the game itself is not crafted as an art. They're crafted purely as an entertaining and fun way of "playing" songs. I admit I don't have any example in mind of that kind of "musical artistic experience" game, but I'm sure they exist and, if they don't, it's theoretically possible.