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Unless you imply that, as Julia also pointed out, nicknames might have nothing to do with the name, but may refer to a physical trait, personality, job,or something randon. (eg someone being called Red because of their hair colour). So, all diminutives are nicknames, but a nickname isn't necessarily a diminutive, but we hardly ever talk about non-name derived nicknames here, more about diminutives.

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But I'm curious as well if there's something else I'm not getting.