Can a cat sprain its tail?

Absolutely, a cat can have a broken tail. It can be broken by being pulled too hard, getting caught in a door, getting caught under the rocking chair, being bitten by another cat (or dog), being hit by a car, etc.

If it is broken, it will hurt them if you touch it (they will cry out), and they may not be able to lift their tail at all. The tail may look very crooked, with a "kink" in it, or it may look perfectly straight.

A broken tail can sometimes damage nerves that will affect the use of the hind legs, or the ability to go to the bathroom, or urine or feces may dribble out.

Some broken tails will heal completely straight, others will heal crooked. If there is nerve damage it can improve over time, but sometimes it does not.