In my opinion, no.
I play games for fun. Don't take that the wrong way, I like games to have a deep, involving story just as much as the next guy, but a game doesn't have to have a story like that for me to see it as "good", nor can that sort of story save an otherwise bad game. It's hard to explain, I would use an analogy using another, similar medium, but almost all of the other forms of media out there are completely story-based, so if the story is good, the movie is good...
I play games for fun. Don't take that the wrong way, I like games to have a deep, involving story just as much as the next guy, but a game doesn't have to have a story like that for me to see it as "good", nor can that sort of story save an otherwise bad game. It's hard to explain, I would use an analogy using another, similar medium, but almost all of the other forms of media out there are completely story-based, so if the story is good, the movie is good...