As always, spoilers for the entire series lie ahead.
It seems Jon Snow may have had a hint about his lineage from a distant relative way back in Season One. This foreshadowing was revealed in the traditional manner: by a nerdy sleuth on Reddit with lots of time on their hands. Since Season Six ended, dedicated fans of the show have had a field day rewatching previous seasons for clues about the current plot developments.
The conversation goes way back to when Jon was a rookie at the Wall. Jon had just learned of Ned Stark's arrest by the Lannisters. He spoke with Aemon, the maester of the Night's Watch. During this conversation, Jon learns that the maester was born Aemon Targaryen. He told Jon the story of when he learned that nearly all the Targaryen descendants had been killed in Robert's Rebellion.
But when Aemon was asking Jon what Ned would do in this situation, he focused on the contrasts between love and duty:
If the day should ever come when your lord father was forced to choose between honor on the one hand and those he loves on the other, what would he do?
We know now that Ned does the right thing by protecting his sister's son by saying it's his bastard. But here's where Aemon revealed he might have known exactly whose Jon's true parents were:
What is honor compared to a woman’s love? And what is duty against the feel of a newborn son in your arms? Or a brother’s smile?
The Tower of Joy scene revealed a newborn son in Ned's arms:
And a brother's smile:
In Season Four, Aemon has a conversation with Sam in which he also says that "love is the death of duty." So, it may just be kind of his catchphrase, but since Aemon is a Targaryen, it's unlikely George R.R. Martin wrote that conversation with Jon by accident. Where and how that plays out in the rest of the show remains to be seen (or until someone figures it out from previous episodes and posts a theory online).