I guess you'd need to define "secret" because what are they referring to...Bigfoot, Jimmy Hoffa, why is Lindsey Graham not married?
If they're referring to the US Gov't, then, yes, I'd be surprised if America didn't safeguard many tomes of secrets. I've read newly-elected presidents succumb to Canities Subita after being briefed on said secrets. I was in the military, stationed overseas, worked on the flightline and clearly briefed on what'd happen to me if our base was overtaken. It's a farfetched scenario but a scenario nonetheless; however, I knew secrets and those secrets had to remain secrets.
The funny thing about secrets is only trusted people know secrets. If somebody reveals a secret, you can't trust them. If you can't trust them, it's probably not a secret.