Followers can sometimes get a glitch that causes their default armor to secretly multiply which in turn stops them being able to take anything once they go over their weight limit. As arrows are weightless they do not count of course.
The only thing you can do is to raise Pickpocket to 100 and take the last perk which allows you to steal equipped armor and simply remove all their armor to reset them which should hopefully remedy the problem.
If you are making your followers loot containers with loads of items with the take command to carry more than they can it can also mess up their carry weight permanently and is near impossible to fix by the way without mods or console commands that only PC's have.
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