I normally don't care about stories in games, but there are some points in some of them that got my goat.
1. FFVIII: Everyone came from the same orphanage....WTF? They kinda made it clear that these guys didn't really know each other that well earlier in the game if I recall correctly, then this happens.
2. Mega Man X6: Zero just randomly comes back from the dead. This is egregious especially because in X5, Zero was cut in half and sent to outer space, kinda like Frieza from DBZ. But then X6 comes out, which apparently takes place mere weeks after X5, and Zero is back. He apparently somehow put himself back together in that time like it was nothing! What the hell?! Sure, he's a robot, but the way Capcom explained the whole thing just made it seem as if he's some type of god or something.
3. This one didn't really bother me, although I got a good laugh from it. Star Ocean 3's whole world is basically a video game, as explained by a plot point that happens waaaaay late into it. So it's basically a video game about a video game.
4. Star Ocean 4: THE ENTIRE. GODDAMN. GAME. Hated it. It's still on my "worst RPGs ever" list. And not just for the story either.
5. Grandia 3: Grandia 3 started out with a young man looking for his hero, an airplane pilot named Captain Schmidt. The game made it seem as if that was the mission for the whole game, but you actually find him at about 3 hours into it. The story was pretty interesting when it was about that. After that the story becomes a typical clustersmurf of cliches and lame plot twists. Sad to say, when this happened, I kinda expected it. It IS a Squenix game, and all.
6. MGS. Not that MGS ever had a good story in the first place; the fact that people are arguing about how crappy the story is are pretty much proving my point.
7. Mega Man Legends games: Taken on their own, the MML games have pretty damn good stories. The problem is that Capcom confirmed that these games take place after the year 8000 (or centuries after the Zero series). I find that really lame and hard to believe, since the technology in the MML games is pretty primitive compared to what's in, say, the X games. Maybe it's supposed to be the world after so many years of advanced technology, and people taking it for granted, which would kind of make sense. Another thing is that by the time the events of Zero happened, the world was overtaken by robots. In MML, the humans are actually artificial humans called Carbons. I don't remember if they really explained how they came about (possibly in the Japanese versions, I can't really recall right now) but it seems to me as if Capcom meant these to be in their own timeline, and just made up the fact later that they take place waaaaay after Zero.