For me, as a BrE speaker, "I've broken my nail" sounds more proper in this situation. However, some people might say "I broke my nail". My feeling is that the latter might be more common in AmE.
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For me, as a BrE speaker, "I've broken my nail" sounds more proper in this situation. However, some people might say "I broke my nail". My feeling is that the latter might be more common in AmE. In any case, "I've broken my nail" cannot be marked wrong.