Is that a joke?
Jack received a Dear John letter from his girlfriend because he has broken her heart.

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It doesn't seems like a joke cause 'Dear John Letter' is a title given to a letter or message which says that ones relaionships  are no longer or do not want to continue the relationship.

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A Dear John Letter is a letter of breaking up.

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It is called a "dear john letter" .. It does not matter what his name is.