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> Are there any use cases where allowing reading t[nil] or t[nan] is useful?

t[nil] is what happens when you do t[x] and x is nil :-)

This may or not be a programmer error: you may be trying to determine
whether a field given by the user exists in a table. If the user does
not define the name of that field, then it's nil and you get t[nil].
Not too common an idiom, though.