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I know for a fact that I didn't feed 200+ berries to gym mons because I ran out after the first 50 and stopped going out to feed Pokemon once I realised that returns on consecutive feeds was absolute shit. It's not bad if I lived on top of a gym/stop cluster and kept feeding something 24/7 for the badge but I live in the sticks so I am 100% sure I didn't feed 201 berries.
So does that mean that the badge counts berries fed to your Pokemon by other people? That's the only other explanation I can come up with that's not "Niantic fucked up counting".