If it's Super Mario Sunshine, then I will quit

Same here and in BF3 case I can say that game is terrible in single player haven't tried the MP so I won't give my opinion on that.

I tend to not leave a game unfinished so I try and push till the end.  Most games will have some redeeming quality, so it might take another shot for the game to click with me.  There are exceptions.  

1. Lair on PS3 - Motion controls were horrible.  Gave up after they released a patch that didn't require the use of motion controls and I still didn't like it.
2. Zack and Wiki on Nintendo Wii - I didn't like this game at all.  Gave up after the third time.
3. Yggdra Union: We'll Never Fight Alone on PSP - I think I was just really bad at it.
4. Genji: Days of the Blade on PS3 - I didn't like this game at all.  Gave up after attempting it twice

Rayman Origins comes to mind...
I never finished Okami, but that's more because I got distracted by other games than because I didn't enjoy it...
I never finished Darksiders for the same reason...
I was halfway through Portal 2 when my computer crashed; I haven't gotten around to reinstalling it...
I'm more likely not to finish a game because I'm playing too many games at once, not because I'm not enjoying them...
I'm currently forcing myself to finish New Super Luigi U...
I think it's unbelievably tedious...

Yeah i hate not finishing games but what happens is if im that bored and not enjoying the game i just get pissed off at it and sell it or delete it if its digital. That way its off my mind.

If I bought the game then I will try to push and hang in there to the end, but if it's just that bad then yeah I'll forfeit. The only two instances of this has been Sonic 06 and Paper Mario Sticker Star, the former of which I knew was bad but fanboyism blinded me, and the latter I just assumed was good because it was a turn based Paper Mario and didn't look up the game otherwise. Wont be having that happening ever again!

First push through, then quit and trade in.

Checking my game collection these are the one I gave up on this gen:

- Zelda Skyward sword: I went back to it 3 times and made it quite far, yet in the end decided I just was not having fun with it anymore due to controls, repetition and lack of overworld content.
- Kingdoms of Amalur: I got very bored halfway through, way too many nonsense side quests.
- SSX: was fun at the start, but I played it far less then any other SSX game. Fun wore off quickly.
- NFS Shift 2 Unleashed: Input dampening lag made the faster cars unplayable.
- NFS pro street: Bought it used for $5, worst $5 ever spent.
- Silent Hill: Homecoming: Did not feel like a Silent hill game at all, zero immersion.
- Assasin's creed 1: What a comeback that series made after the first installment.
- Lair: It wasn't the controls that made me give up, the chaotic mission design interrupting you constantly did it.
- Folklore: Tried to get into it 3 times, I was simply bored.
- Virtua fighter 5: I gave it a chance, fighters are not my cup of tea.
- Ace Combat 6: Every mission felt like the same long drawn out procession.
- Forza 2: I unlocked all the tracks in arcade mode, didn't like enough of them to do career mode.
- We Skii: Bought it for the balance board, big mistake.
- Harry potter Order of the phoenix: Did I really buy that... apparently I did :/
- DKCR: Bought it to play together, however co-op sucked.
- SMBB: Even with cute Nintendo characters, fighters still don't interest me.

It seems like a lot, but not too bad out of 147 console games last gen. (not counting psn/xbla)
There's still a bunch of 'will finish later' games, not given up, just not enough time to play them all.

Quitting on FPS/3PS games is far less likely with 8-12 hour campaigns. I spend more time then that on some of the games I gave up on. Sometimes a short campaign is a blessing.

I always finish a game that interests me, and don't bother with boring games altogether.
The one time I broke this rule was during World At War. I hated how many missions were based on surviving waves after waves of enemies. But I carried through just so I could hear Reznoves epic speeches again.

Depends on how much I payed for it. If I put down $20+ on a game, you can bet your ass I'll finish it. No matter how awful it may be.

Depends. If I don't like the gameplay then no, I won't finish it. If I don't like the story but the gameplay is more than decent then ill push through.