Its possible,and you can always think this is what is happening when you play your characters,but I personally doubt it - the Nerevarine is apparently alive and in Akavir by the time of Oblivion(could be a conspiracy though),and the CoC should be Sheogorath (or something similar) by the time of Skyrim (we need more information on this,but you could make up any story you wanted if you really wanted to play as the reincarnation of the CoC).Also,the Dovahkin in Skyrim is phrophised (can't spell) almost as "the one and only".While this may be because he is the last of a long line of Dragonborns it doesn't sound like that in the phrophecy.Still,you could imagine that your character reincarnates into a different body,spirit and role each game,but I don't as it sounds cliche for a hero reappearing "In a time of darkness".But,its up to you reallly.
In my fanfic story, my character Vedaa returns back to her home in Summerset Isle, and so the rumor that the Nerevarine went to Akavir is just that, a rumor. However, everybody believes the rumor to be true and thus, nobody believes my character is the Nerevarine, which leaves her as just another dark elf once again. She doesn't do the Oblivion story, and so I just use a random hero to fill in the role of the CoC. Now my concern with TESV's storyline is that can the Nerevarine also be the Dovahkiin? I don't see why not but I just want to see what other people think. My character was created by the Daedra & Aedra to fulfill the Nerevarine prophecy. But since the emergence of the TESV story, would it be plausible to have her creation be for the purpose to fulfill both the Nerevarine prophecy and the Dragonborn prophecy? TESIII and TESV are centered around a prophecy character. But the CoC wasn't a prophesized hero, was he? His face was in Emperor Uriel Septim's dreams, but that doesn't necessarily imply a prophecy. Perhaps it could have been a visual message from the gods that they have selected a unique individual who will help stop the Oblivion invasion. Basically, I'm just trying to patch holes to make my story make sense and be lore-correct.