The Nerevarine was called "Dragonborn", but that could have had a different connotation, such as "Born under the Dragon", i.e., born under the Empire. We can assume that the Champion of Cyrodiil was NOT Dragonborn, because he was unable to wear the Amulet of Kings.
If it's a matter of lineage, that's fine, but your ancestors would have known they had it. From what it sounds like, being Dragonborn isn't just something that you just randomly discover. It's a really, really big deal.