What is the criteria for feeding berries?

I like feeding Pokemon with berries on gyms. However, it is not always allowed to feed them.

I already know that you cant feed more than 10 in a certain period of time. I also know that you can't keep feeding the same Pokemon over and over again. what are the limits for feeding berries to Pokemon in gyms?

The limits, for one trainer:

• 10 berries per Pokémon per half hour, fixed window (so it resets after 30 minutes after the first Pokémon was fed)
• Any number of Berries to ten unique Pokémon per half hour, rolling window (so each time it's been 30 minutes since you fed a Pokémon, a new one is eligible)

As you feed a Pokémon, each successive berry after the first restores less health; this is reset after thirty minutes as well (with the fixed window).

There are no known limits per Pokémon for multiple trainers: I have seen spoofed gyms being defended prior to legendary or EX raids where clearly dozens of bot-trainers are feeding berries to the same Blissey.

See this answer by Vemonus and the similarly linked Silph Road thread for more details.