I think the lowest ranked character that ever won a large tournament in 07 was ICs for Pound 2 but that's not really fair because Chu is a top level player, and easily one of the top 5 players for that era. Mango at Pound 3 was also similar, with mid tier / high tier (whatever you count Puff as) winning it, but again it's troublesome to use that because Mango's really good so you could also make the argument for simple player skill. Also, M2K hates Puff
On West Coast there was a large tournament with Luigi Ka-Master and Hugs in the finals, so if we're going off slightly smaller (but still formidable) tournaments, then Samus is also viable, but again we have the issue of Hugs and Ka simply being better than people at that time. Hugs is semi-retired from Melee (or did he quit entirely? Someone correct me on this), and Ka doesn't travel much, though, so - like Pound 2 - it's somewhat of an isolated incident.
The top four have a significantly higher amount of winning power than the characters below them. I think Jigglypuff, Falcon, and Peach all have enough to win good sized events, and get far, and a good pilot can take a mid tier pretty far, but whether or not the mid tiers can actually pull off a win with how gay Melee's become / is becoming is a tough one. I'm optimistic, though, so I'll give a very guarded "yes".
For major tournament, I think the upper end of mid tier can go far but I don't think unless they're more skilled or lucky or something they'll be able to win. Similarly, I think it would be brutally hard, but high tier could probably do it too.