Most games have one beginning, one ending/conclusion and progress in a linear fashion from the beginning to the end (with bubbles of freedom of action along the way).
It depends on what you actually call the 'Game'.
Let's take Starcraft 2 as an example. The 'Game' is an archetypal RTS with resource mining, army building etc. The 'Story' is the single player campaign, which is essentially a string of 'Games' with a bit of extra scripting, with a bit of exposition in the form of cutscenes holding it altogether. There are also some minor interactive components here, like the ability to choose which map to do next, or which research upgrades to buy or whatever, but nothing that really makes any difference to anything.
Playing through this kind of 'Story' is a bit like watching a short and slightly disjointed movie which pauses every so often and requires you to win a game to continue. If you're lucky, some chapters of the story you can choose to watch in a different order. Or to look at it another way, it's like playing a sequence of scripted single player maps, except between each one you have to sit through a bunch of cut scenes.
The success of this formula depends on a lot of factors. The 'Story' will have to make some concessions to justify the sequence of games within it, and to avoid annoying the player with over long cut scenes and exposition. These concessions can render the narrative irrelevant.
The 'Game' will have to make some concessions (usually via the scripting) to the story, often in the form of scripted events, limitations, and special victory conditions. These concessions can sometimes lead to some really enjoyable scenarios. However, they can also utterly destroy the gameplay, paring the options back to such a degree that you're just clicking your way through a linear script.
In short, I don't think this is a particularly great way to incorporate story into games. It often suffers from exactly the problem I originally spoke about, where both the story and the gameplay can suffer as a result of their combination. For me, this model only works if the gameplay remains intact - as a general rule, if the gameplay is there, I couldn't care less if the story is rubbish, so long as it's skippable. Sacrificing gameplay for story on the other hand, is a big no-no.
I believe much of the success of the Starcraft-franchise wasn't just about the actual game-mechanics. What got me exited about Starcraft 1 was much about the whole tone/setting/atmosphere of the game. Starcraft 1 did this nicely, a few alien-esque cut-scenes, a pretty well-written and cool back-story (if you bothered to read it in the game manual), and a simple story goal of unifying against a common threat. Everything in the game mechanics, the gory deaths, the voice acting, music, etc. played into the narrative. We have a game where the game-mechanics playing into the story and vice versa.
If you removed those story-elements and just have 3 random races battling each other with no context or feeling of anything being at stake (the human race), I dare to say it would not be as successful of a game.
Again I think the idea of story "getting in the way" is more indicative of bad writing than that having a linear narrative and game-mechanics does not mix. Look at the most loved games of all-time and what they have in common: Chrono Trigger, Mass Effect, Half-life 2, Final Fantasy VII, Metal Gear Solid, Portal, Legend of Zelda: Ocarina of Time, etc. They all have a strong cohesive, unified, engaging (linear) narrative as well as compelling game mechanics (and they go together).
I'm not saying there is no place for games where the players makes their own story - in fact I think there is a place for it. I believe it can be done in MMOs and multiplayer games - you can have a place where players play a role in economical forces, power struggles, diplomacy, etc. set it up right and intrigues and stories will emerge just as in real life. I would love to see a game like that.