During Prisoner of Azkaban, Harry and Hermione use the Time-Turner to save Buckbean and Sirius. Snape suspects Harry was involved in Sirius’s disappearance. To calm Snape, Dumbledore points out that they couldn't have helped him unless they were in several places at once:
“THEY HELPED HIM ESCAPE, I KNOW IT!” Snape howled, pointing at Harry and Hermione. His face was twisted; spit was flying from his mouth.
“Calm down, man!” Fudge barked. “You’re talking nonsense!”
“YOU DON’T KNOW POTTER!” shrieked Snape. “HE DID IT, I KNOW HE DID IT—”
“That will do, Severus,” said Dumbledore quietly. “Think about what you are saying. This door has been locked since I left the ward ten minutes ago. Madam Pomfrey, have these students left their beds?”
“Of course not!” said Madam Pomfrey, bristling. “I would have heard them!”
“Well, there you have it, Severus,” said Dumbledore calmly. “Unless you are suggesting that Harry and Hermione are able to be in two places at once, I’m afraid I don’t see any point in troubling them further.”
— Prisoner of Azkaban, chapter 22 (Owl Post Again)
Of course, they can, and Hermione has been doing it all year for her classes.
Were Snape (and the other teachers) unaware that Hermione had a Time-Turner? If they weren’t, then surely Fudge was. McGonagall had to write to the Ministry to discuss it:
“It’s called a Time-Turner,” Hermione whispered, “and I got it from Professor McGonagall on our first day back. I’ve been using it all year to get to all my lessons. Professor McGonagall made me swear I wouldn’t tell anyone. She had to write all sorts of letters to the Ministry of Magic so I could have one.
— Prisoner of Azkaban, chapter 21 (Hermione’s Secret)
Surely for anyone who had knowledge of the existence of a Time-Turner, whether they knew Hermione had one or not, Dumbledore's logic would be unsatisfactory?