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A lot of those games never focused on story. I could say Bejewled by that standard and call the thread finished.

Of games that tried, I would say the worst I have experienced off the top of my head would probably be FEAR 3. It's a mish-mash of nonsense and fucks over the previous two games as well, which while not Shakespeare were written pretty well for FPS games.

I vaguely remember Pariah, a shitty Xbox era shooter, having reviewers call it out for a poorly done story in particular.

On the RPG front I think Oblivion's main plot and dialog are spectacularly bad for that kind of game, brief quality like the "paradise" section aside.

I honestly think the COD games post 3 have the most convoluted stories in any video game I have ever played.

Oh man, forgot these... Modern Warfare was pretty good actually, story wise, but Modern Warfare 2 is a total clusterfuck.

The storyline in Max Payne 3 was pretty bad. And it's a rockstar game too!

Pong. =P

Tales of the Abyss, for me. It's horribly remarkable that I played all 50 fucking hours of that game, and still didn't know what a phonon was. In fact, I didn't know what the hell was going on 95% of the time. And it wasn't that I wasn't paying attention, I was glued to that stupid screen, and watched every optional dialogue available. I think it's just that its plot was garbage, its characters unlikable (never saw the appeal for Jade, and Luke was always intolerable; although, Tear was nice), and the dialogue + voice acting atrocious. Aside from the story, the the art design was bland, the gameplay was ok, but repetitive, and the whole game was 20 hours of filler too long. Screw that game, and screw the 40 dollars I payed for it :(

I'll try to remember other bad storylines in VGs, in the meantime, I'll try to give excuses and comments as to why those ridiculous storylines make sense.

10. Double Dragon.

My guess is that cops back then were to fat to jump over pitfalls and dodge spikes; and their bullets lacked the sheer awesome power of Billy and Jimmy's martial arts, that could even fight a guy with a gun even after taking a few shots to the gut.

9. SF II: TWW.

Isn't it standard proceedure for all those that fight evil organizations to street fight all over the world until they find their target?

8. Shatterhand.

This storyline proves that is something lacks power, replace it with it's metal version.

7. Cheetahmen.

These guys have no excuse, they could learn a thing or two from Captain N.

6. Rampage.

That's why I don't trust those scientists, too much risk of having them turn you into giant classical movie monsters.

5. Contra.

Well, they are shirtless most of the time, and everybody knows, as proven by Rambo, that a shirtless man equals an entire army division.

4. Battletoads and Double Dragon.

Hey, if I were a group of space toads looking for Earth's mightiest warriors, I'd definitely pick two brothers that decimated an entire whip wielding, steroid using gang with nothing but the power of fists... and kicks.

3. Bad Dudes.

These guys have no excuse either, they are pretty easy to manipulate: "are you bad enough dudes to pick up the groceries, oh, and some gum on the way back?!"

2. Blaster Master.

I'm pretty sure this was the draft for some 80's anime, which almost always started with a boy finding an abandoned and fully equipped military vehicle.

1. Super Mario Bros.

You know what they say, it's awfully suspicious when a princess gets kidnapped over and over and over again, in fact, it's so disturbing I'm not even going to give hints *cough* bowser suddenly has children and she invites him to parties and kart races *cough*. On the other hand, what if she is actually in cahoots with bowser and tries to lure Mario into enemy territory with her so called "kidnappings".

Wha? I loved it.

http://tvtropes.org/pmwiki/pmwiki.php/Main/RuleOfCool

As Stinging Velvet already mentioned:
Some videogames don't need a story.
Tetris?

Okay, but what about Lode Runner? I loved this game but I couldn't figure out the story one bit.

The worst stories in games I played in recent years are driving games. Somebody stole your ride? Get this junk of a car and race until you can challenge him or whatever plot twists I have long forgotten.

Shooters really have bad stories too, Doom, Quake and Wolfenstein were all pretty bland. I really like the Unreal Tournament story. You really feel like a champion when winning a final tier match there.

After years and years I still have no idea what the hell happened in the last quarter of Fahreineit/Indigo Prophecy

Okay, so let's pick something that isn't from the 'Ignore the story, this game is just supposed to be fun' box.

Skyward Sword. This is the most convoluted story that I've been unable to swallow. I'd love to go in depth on the issues, but I'd rather not drop an entire paragraph here.

Metal Gear Solid 2

Retro City Rampage (probably intentionally)

Except that that's probably not your list. You could at least have tried to make an effort.