The story's main character has a story-breaking secret. When are you interested instead of annoyed?

You will have to decide your ownself which secrets are and are not storybreaking. Or, of course, we can argue about it.

Open spoilers for Suits below (and other shows, stories, and books if the thread gets any legs) so be warned.

Mike Ross, from the aforementioned USA network series , has a genuinely story-breaking secret. He’d be unable to keep his job if it became common knowledge that
he never passed the bar, attended law school, or even graduated college, and thus is not entitled to act as an attorney. His mentor Harvey likewise is likely to lose his license if that gets out. I like the story but this aspect bothers me because it’s unnecessary. Mike could just as easily be a super-genius who had passed the bar but pissed away his first big law job without having to be an utter fraud. I just can’t believe that Harvey – or, worse, Harvey’s boss, Jessica – would take the risk. They can dance around revealing the secret but we know they’ll never reveal it.

Anybody else have a suggestion?

Hypothetical: Gilligan had a spare shortwave radio in his back pocket all along.

Real cases that are annoying: Character X is secretly working for the bad guys, yet before this is revealed, Character X has been instrumental in helping the good guys. Why is X helping the good guys, when his/her motivation should be to misdirect or delay them? Who knows. Example: Valeris, Star Trek VI.

Just a clarification, Mike Ross has passed the bar, repeatedly, just not under his own identity. Passing the bar for others was how he was making at least part of his living before Harvey hired him.

Three’s Company had the secret that Jack was straight, if Mr. Roper or Mr. Ferley found out he’d be evicted.

I would say Bosom Buddies, but it did come out that Buffy and Hildegard are really Kipp and Henry, yet it wound up not mattering for some reason.

3rd Rock and Mork & Mindy had the same secret, that the characters were aliens, if it came out, hunted, studied, vivisected, etc. ALF was similar, but since he didn’t look human it was more about being harbored by humans than passing himself off as one like the others. I’m sure many more can fit here.
edit: IIRC, Valeris helped them at times because her group was playing two sides (Fed/Klingons) against each other. Sometimes her goals were met by helping, other times by hindering.

I don’t think Valeris really qualifies. For one thing she wasn’t the main character; for another, exposing her secret didn’t break the story, but was a requirement of the plot. Anyway, her behavior while Kirk & McCoy were prisoners doesn’t seem all that poorly thought out to me. Her job was to help foment a war between Federation & Empire, not to see that Kirk & McCoy died on the prison planet. Having the Enterprise rescue them (and thus, from the Klingons’ point of view, demonstrate complete lack of respect for law) could only help matters.

Glinda: You don’t need to be helped any longer. You’ve always had the power to go back to Kansas.
Dorothy: I have?
**Scarecrow: **Then why didn’t you tell her before?
Glinda: Because she wouldn’t have believed me. She had to learn it for herself.

I’m not going to spoiler this because it occurs in the fracking pilot.

On Battlestar Galactica if the traumatized humans knew Baltar had unwittingly handed them over for the slaughter to his hot blonde babe they probably would have airlocked him as fast as possible(ditto for Balter being guided by Cylon angels) and the series would have ended with the whole fleet getting creamed.

I liked it, ALTHOUGH I thought the contortions near the end to keep the secret got annoying.

Spaced had something like that. The main characters have to pretend to be a couple in order to keep the place they’ve rented. Although eventually it does come out.

It turns out that the ‘couple’ requirement never existed (error on the newspaper’s part) but the landlady is upset at being lied to regardless.

Lion: So why did you just t-t-t-t-t-tell her?
Glinda: All right, it’s all just a dream. And you’re about to wake up anyway, so I wanted you to have an excuse. Some people just have no knack for storytelling conventions.

Gandalf, at the Council of Elrond: “Or, we could just have Gwaihir fly Frodo to the volcano and drop him in. Er, the ring. Drop the ring in.”

I hate secret crap. I don’t mind if it builds up a little bit, but the answers had better be forthcoming toot-sweet! The Unusuals is a good example of how to make secrets keep things interesting; Lost is the opposite. I hated Lost the moment I saw the first commercials; think I watched one episode and said ‘o frak this crap for a load of shite’. And I was right!

To be clear, I’m not talking about things that are kept secret from the audience. I’m talking about things that hte audience knows about a character which that character is keeping from most or all other persons in the fictional universe. Things like the fact that Gibbs on NCIS murdered the man who killed his family, or that Don Draper’s real name is Dick Whitman. Admittedly the second isn’t necessarily a story-breaker if exposed, but the first sure is.

D’oh! Sorry, Skald!