Your reasoning can be applied to a regular job: if I am tired I make more mistakes and every mistake has a cost. If the cost of mistakes made during the extra hours exceed the value of the job done, those extra hours didn't produce anything useful.
Even not taking into account mistakes, there is another important factor to consider: you need to recover energy from one day to another. The more you work the more you need to recover and less time you have to do it. If you don't, you will be less productive the following day.
I think that for this reason it is reasonable to say that there is a limit after which the productivity start to drop, though it probably depends on the individual and on a number of factors.
Even not taking into account mistakes, there is another important factor to consider: you need to recover energy from one day to another. The more you work the more you need to recover and less time you have to do it. If you don't, you will be less productive the following day.
I think that for this reason it is reasonable to say that there is a limit after which the productivity start to drop, though it probably depends on the individual and on a number of factors.