I wouldn't say I've seen any consistently weak civs, I think it has more to do with starting position.
The only three things I really know is that Montezuma and Nobunaga are warmongers and Gandhi isn't.
I think I agree with this. Even though we say that some civs are weak in the hands of the AI, I've seen just about every civ become a runaway at some point, even the weakest.
I've seen a runaway Aztecs once. And a runaway Mongolia, in one of those lucky (for them) games where they were attacking city-states and civs left and right and like a Karma Houdini, somehow didn't have the entire world come down upon them. Mongolia had conquered two CS, and since I was Brazil that game and going for diplomatic victory, it irritated me, but I didn't want to attack him since he was the only one who shared my ideology. And I needed him as a buffer against our ideological opponents, Assyria and Rome. Let's just say I was paying Genghis lots of money to make peace whenever he attacked a CS. Oh, and Rome was a runaway that game too, having swallowed up both Germany and America.
From what I've seen, I think the times Rome does poorly are the times they pick Honor as a starting policy, which happens pretty often.
As for India, well...there was the time I played on a large Earth map, and India started where the central US would be, along the Mississippi River. And then got Hanging Gardens. And a ton of other wonders. And then they went Patronage and started allying every CS in the world. The Aztecs to the south even took one of India's Central American cities during the Renaissance, and it didn't even slow India down. Then it turned out that India had lots of uranium, so of course they went ahead and built 8 Atom Bombs. I was Russia in that game, and I could barely match that nuke stockpile (as I was building Nuclear Plants as well). There was also a runaway Iroquois in that game, but runaway India easily exceeded his score.
The hilarious thing was that India wasn't alone in North America, they had Siam with them to the north. Poor Siam, starting in Canada. They actually managed to do pretty well considering how bad their start was, colonizing Alaska and the rest of Canada.
And it must be a testament to how peaceful Gandhi is that he didn't wipe Siam out, didn't go to war with them once.
Tl;dr: Even the weakest civs in AI hands can become runaways. The only thing I think I've never seen is a runaway Songhai. Even got their capital conquered by a city-state once...
