They can be winks to the reader without the character being aware of it

I’ve always felt he knew, or at least suspected, Jon was something else

His talk with him about the Targs, and the whole ‘A Targ alone is a terrible thing’ have always felt like winks… I don’t think he knows, but I think the thought has crossed his mind.

Bloodraven could probably plant a dream or two.

How would Aemon know? Plus if he did he wouldn't be so adamant that Dany is the prince who was promised. He'd consider Jon too

Cat literally betrays King Rob letting Jamie go

Ned made the right call.

Only a few folk know. Ned, Howland Reed, and Bloodraven do… maybe Benjin / Aemon. No one else.

The Cat-Ned relationship wasn't always as strong as it was. Cat expected she would marry Brandon, not Ned, and they were effectively strangers when they got married. A younger Ned wouldn't have known for sure that Cat would never betray him, that trust built up over a decade of marriage.

So he wouldn't have told her right away. I feel like he could have told her later on, though.

Cat is generally a dumb person who makes dumb decisions, she probably would have told Lysa who would have told Littlefinger who would have started at least 3 wars with that intel. Ned did good to not tell her and ordering her as her lord to get over her feelings and that his bastard was staying with them.

No he doesn't. He would've said something about it in his semi conscious ramblings just before he died when he talks about Rhaegar, Daenerys and the Prophecy. Also Rhaegar died before Jon was born, so he would've had to tell Aemon in advance

Do you think Aemon knows? I’ve always got the impression he did

Ned strangled the woman who helped with the birth with his shoelaces immediately afterwards though

There's at least 3 people who know the secret. Ned, Howland Reed and at least one woman to help with the birth. There's the possibility that Benjen knows too. That's 4-5 people already. What's that saying. It's too nuclear a secret to tell to anyone who does not have to absolutely know

……or kept the secret knowing it would put her family at risk if it got out but harbored a resentment for him

This right here. Cat is fanatical about protecting her family and harboring the last Targ, born from the woman Bobby B started a war to get back and clearly never got over would put her and her kids in the line of fire. Actual danger, not just her pride and feelings being hurt at Ned being publicly unfaithful. If she’d known the truth she would have yeeted that baby without a second thought.

"Hey, as a wedding present, want to be party to some light treason that could get us all killed if my best bro Bobby B ever finds out?"

At first, they don't really know each other. Besides, it would have made her an accomplice of his high treason towards Robert, hence exposing her and their kids to a harsh sentence (not that I believe such a little detail as Cat not knowing would have stopped Robert, but Ned may have believed it).

Then... Ned loves Cat, and she loves him, but he knows she'll always put her children first in case of necessity. I think the explanation of Ned's silence is in the moment he thinks about Jaime pushing Bran to save his own children.

>Ned thought, If it came to that, the life of some child I did not know, against Robb and Sansa and Arya and Bran and Rickon, what would I do? **Even more so, what would Catelyn do, if it were Jon's life, against the children of her body?** **He did not know. He prayed he never would.** 

And as much as I love Catelyn (who's one of my favourite characters, because she's real in a way few mother characters are in fiction), I think we all know what Cat would have 

He married a total stranger, he probably thought it best to not tell people. It's not a secret once people find out.

This is a secret that if it ever gets out then anyone compliant could be executed as traitors. Keeping Cat in the dark was as much for her protection as Jons. If his parentage ever came out at least she could have deniability

Well, it would kind of be like Sansha keeping a secret except Catelyn wouldn't have been calculating about it.  Lol

TLDR; he didn't know her "like that" yet. 

my brothers and I have a love/hate relationship. My eldest brother, in particular, traveled five hours the day of my son's birth and was present up until I delivered. He even pointed out my low blood pressure to the nurses ahead of them realizing there was trouble. 

Lyanna and Ned shared a similar sibling bond. While Ned genuinely grew to love Catelyn, there's no way he was about to endanger his sister's child. He was the only thing left of her and you know every time he saw Jon, he saw Lyanna. I think having to see his sister's face on her son re enforced the promise every single day.

She would have let the Cat out of the bag.  Pun intended.

Ofcourse she wouldnt have betrayed Ned..
But thats not how you keep secrets, you keep them by not telling them.

In the first book Ned gets distracted very soon with issues and doesn't have the time to think about Jon except for the time he is in the cripts. He also leaves early halfway through the book.
But it would be impossible for Cat to not think about Jon's parentage even once during the course of 3 books.  And that would reveal the mystery.

Why didn’t Ned tell Catelyn about Jon. She could’ve gone along with the story too. I don’t think she would’ve ever betrayed Ned.