It's not really clear what you mean by the brain accumulating noise. An analog computer can suffer from compounding imprecision, because error bars carry forward. We are analog computers in a vague sense, but as humans we aren't imprecisely crunching numbers all day such that our results are out of whack by nightfall... You could say sleep is like a reset for the brain, but that doesn't say much. Sleep is complex and relates to most everything else about an organism in complex ways.

Nah, humans just blank-out for a couple of seconds mid-task instead.