Do Cats Get 24-hour Flu?

Trouble has been doing a whole lot better since he started his medicine. His appetite is back, his energy level is back up, etc. But on Saturday, he didn't leave his kitty bed except to go to the bathroom twice in rapid succession, both times insisting on going outdoors rather than using the litter box. He didn't eat all day, either. In the evening, I hand-fed him some kitty treats, and he ate four of them, but refused the fifth. About 15 minutes later he vomited them up.

This morning, he's awake and starving and begged to be fed and has eaten 3/4 of a small can of wet food. He seems to be back to normal.

Do cats get 24-hour flu?