Let me put it this way. You show me any game from the 80s or earlier that has a good story. Any game...AJey said:WTF? Jim, you serious? "Game can be good with a weak story"! Are you F-ing serious?! Most pathetic piece of argument ever! wow! Show me a good game that has a weak story! You gonna use Angry Birds maybe?! Show me a single good game without or with weak story!
Now think of any game released during that time that were actually good: Mario 1 and 2, Megaman 1 and 2, Street Fighter 1 and 2, Wolfestein, Pacman, Galaga, Tetris, etc. Any game in that group is an example of games that are good but have almost no story. And that not even counting more modern games.
Story is games is almost always weak... Other than RPG, most games stories is more like a premise (save your girlfriend, get the MacGuffin, shoot everyone before they shoot you). Most modern games focus on setting to set them apart, more than story.