After re-watching season one episode two I believe Jon Snow really was Ned Stark's bastard. When Catelyn Stark was sitting by Bran's bed after his fall and Ned was about to leave for the kingdom she brought up how seventeen years ago he left her and brought back Jon. You could see It all over Ned's face that he had slipped up.
Then when Ned and Jon were about to part ways Ned told him straight out flat Jon you're a starks you have MY blood in you. He didn't say you have Stark's blood but MY blood. The when Ned and the King were talking about Jon's mother Ned got really heated because he had made that mistake and didnt want to talk about it. Not because Jon had Targaryen blood. Ned really didn't seem to give a fuck about the Targaryens. Ned was out there getting his dick wet and got caught up.
