Even though nobody's perfect, there's a weird thing in American business culture of never admitting mistakes.
The usual reason is that admitting a mistake could open the doors to a lawsuit--but this is bunk, as has been shown in medical research (tl;dr version: admitting mistakes and engaging in honest communication with patients cut malpractice costs by over 50%).
The truth is that most people and companies just don't like to admit mistakes. It's hard, it's embarassing, and if the mistake was highly visible or expensive, it can feel like an end of the world moment.
But from the other side--the perspective of the victim of the error--things look completely different. Admitting a mistake (especially when the stakes are high) makes you look responsible, honest, and trustworthy.
Mistakes are also opportunities. As the old saying goes (I heard it in flight school), "Experience is just a series of nonfatal mistakes." Taking ownership of a mistake empowers you to learn, change processes, and improve. Even taking ownership of someone else's mistake can be highly empowering: often there are things you can change to make a mistake less likely or less damaging.
When the excrement hits the whirling blades, it's natural to hope it wasn't your mistake. And it's natural to be relieved if it wasn't. That's my reaction, too.
But if you screwed up, admit it. It hurts, but it's the best way to turn a mistake into experience.