This is the basis of a short story by Stanislaw Lem, where a man who had his entire body replaced by prostheses (including first one, then the other hemisphere of the brain) is sued for nonpayment by the prosthesis producer, who argues that since there is not a single organic part left in the man, he's just an inanimate collection of prostheses which lawfully belongs to the company.
There is a story by Stanislaw Lem that brings that question to a whole new level. Unfortunately I can't find that short story right now, but this abstract should give you a pretty good idea:
Now the question is: Is he just a machine because all his natural bodyparts have been replaced, even though he has the same personality etc as the original person he was before, or is he a man who should still be considered a human, even though all his bodyparts, including his whole brain, are now artificial?
Personally, I think to better answer this question one has to assign a "persona" to the objects in question.
Take Redundant's example for instance, and this time let's say that the human in question is named John.
Instead of asking if John is human, I think a better question would be: Is the end product of all those replacements still John?
Some would argue that the end product is still John, He remained John after the first step and thus must be so after every step.
Others however would claim that, since none of the original 'John' remains, the end product is merely a "collection of prostheses."
A third, more extreme, argument would be that even after the initial replacement, the end product is no longer John.
Alex Murphy: disassembled and assembled