Board Thread:TV Show Discussion/@comment-29740399-20161026202030/@comment-28895263-20161027051720

If Ned told Cat that Jon was Lyanna's son, she'd figure out who the father was. Everybody knew that Lyanna and Rhaegar went south together. (Just not whether it was a kidnapping, or an affair.)

So Ned's decision was "tell all" vs. "tell nothing". I'd guess that the deathbed promise Lyanna extracted from Ned was to never reveal Jon's father.

Yes, this was rough on both Cat, and Jon. The best way to protect a secret is to tell no one; Ned stuck to that policy. The secret survived Ned's death, and Cat's death. (Jon's death, too, but that was temporary.)

Cat's hostility to "Ned's bastard" helped to make the cover story seamless.