Well, it's a question which could be turned into a coherent question in a couple ways, so before getting an answer, you need to decide what question you're asking and what an answer ought to look like. For example:

• You could ask whether people can distinguish between biased dice down to single percent level or smaller by rolling them a ton of times.
• You could ask whether calibrated experts can be calibrated down to sub-percent levels without resorting to explicit models and calculation, or whether the inherent mental noise overwhelms differentials before then.
• You could try to tie it to pulse-coding for utility/rewards (lukeprog covered in one of his neuroscience posts) which would imply something like nothing finer than 1/1000th or something. And so on.

I don't know the answers to any of these - my own impression is that people have fairly granular probabilities. I don't bother with single-percent differences in my own predictions on PredictionBook.com unless I'm in the 0-10/90-100% decile (where 0% is quite different from 1%).