The point of the word ‘cycle’, used in relation to ‘REM sleep’, is that it occurs a number of times during a full night’s sleep. It takes place around 60 to 90 minutes into each deep sleep cycle, so approximately 3 or 4 times during the night.
The earlier in the night, the shorter the period in REM, with the last cycle containing the longest, which is how it is possible to recall dreams. Individual dream-sleep periods rarely last for more than 20 minutes or so (I disagree with @Male on this, I believe he may be referring to the total, as @Wundayatta does).
I love that you refer to a cat @Aethelflaed! We have two, they are Bengal-types, with very wild attributes, though well socialised. Cats naturally sleep for many more hours than humans. Watching ours, I’d say the gaps between dream cycles can be anything from 15 minutes, to an hour or so, with the REM lasting 6 or seven minutes. Then they often wake for a moment, stretch, then go to sleep again.
I’m quite interested to know what they dream of, as ours both growl and move a lot as they dream. I know I should get out more…