I finished all the books, and am re-reading A Game of Thrones right now. I've discovered something in the text that leads me to question whether Jon Snow is actually Ned's son.
Speculative SPOILER: ( In the book, Ned and Robert are talking about Tywinn Lannister's treachery in taking the Red Castle, and as an off comment, it mentions that Ned had been living a lie the last 14 years -
"Treachery is a coin that the Tragaryen's knew well," Robert said. The anger was building in him again. "Lannister payed them back in kind. It was no less than they deserved. I shall not trouble my sleep over it!" Whatever lie Eddard has been living for fourteen years, it has to do with a certain 14 year old boy (Jon Snow). What else happened 14 years ago? The death of Lyanna. I suspect that Lyanna is the true mother of Jon Snow, making him a bastard of Raegar Tygaryen, not Eddard. )
"You were not there," Ned said, bitterness in his voice. Troubled sleep was no stranger to him. He had lived his lies for fourteen years yet they still haunted him at night. "There was no honor in that conquest."