An interesting idea. First, let's imagine that it's possible to make a robot that is exactly like me: it has the same mental and physical aptitudes, the same personality, it looks like me and it sounds like me. It would not actually be me: it would exist separately from me, like an identical twin. I could have a conversation with it. If we accept that the robot is conscious it will have it's own feeling of "me".
Now, imagine I am progressively replaced by robotic parts. We might start with limbs, then go on to body and internal organs, ending with the brain, which is slowly replaced with modules. At the end we will have a robot identical to the first one, but with one very important difference: this robot will actually be me. I will continue thinking, moving and talking with my robotic parts and electronic brain, keeping my sensation of being "me".
Could this work? We don't yet have the technology to do this, but I can't see a reason why it shouldn't be possible to develop it. Could I go on existing as a robot, or is there some indivisible part of the brain where "me" resides?