It's very game-specific, so the only sure way to know is either to discuss it in the forums or submit a run and see how the audience reacts. But generally, the allowable circumstances are very narrow. My guess is that most likely a "clean" run would have to already be published and the second run would need to do something
really cool or have a
huge time savings to qualify.
It does happen sometimes with unpublished games, though. Recently, Baxter had his
Biker Mice From Mars accepted with unbelievable audience acclaim. This run uses a password to skip to the last level of tracks and give his bike some powerups; it was agreed that playing through the whole game would have been terribly boring to watch. Yet, because Baxter so totally pwns the game, the reception was overwhelmingly positive. So it can happen.
If you're going to start a run from an SRAM or a savestate or using a password, make sure to
- Discuss it in the forums, and
- do something very impressive in your run.