Do Fish Ever Stop Swimming?

The short answer is Yes, they do, Hence the reason they need to rest to regain strength. The creatures that live in the pelagic environment never stop swimming. It's the same with fish.

Is it normal for fish to stop swimming?

One common cause is improper water temperature. If your fish's water is too hot or too cold, they will be very inactive. Check your heater and verify that your aquarium is at the right level. Other possible causes are overfeeding and improper water quality.

Do fish have to swim all the time?

Some fish, like tuna and some sharks, have to swim all the time so that they can breathe. Its likely that these fish sleep with half their brain at a time, just like dolphins.