Two people have asked essentially the same thing here, I'll answer this one because it was at the top. I am going to deliberately use a far-fetched scenario rather than invoke specific technologies, because there is no benefit to learning more specific here than "Nope, a competent CA must never allow this to happen".
Alice has public key A, and Bob has public key B, and everybody trusts Charlie the Certificate Agency to issue certificates, Alice has one binding (Alice,A) and Bob has one binding (Bob,B).
Alice controls a missile she will only accept Major Tom's commands. Bob runs firework displays, he accepts the display organizer's commands.
I want to fire Alice's missile. I impersonate Bob, and I trick Charlie into issuing a certificate (Bob,A) because she doesn't verify that I know the Private Key for A. Then I offer (as Bob) to run a really great firework show for Tom, and I give him the (Bob,A) certificate so he can command firework launching.
When Tom sends me a launch message encrypted to A, I simply deliver it to Alice, who launches the missile as I desired.
Alice was never compromised, neither was Tom. Charlie's only mistake was not verifying that I controlled the Private Key for the cert she issued me. Bob was compromised, but he was just running a firework business, he didn't know this was a matter of national security.