You might receive one of these when you are getting dumped by your erstwhile partner. I don't think it is a coincidence that the a recent movie called Dear John includes a Dear John letter as one of its plot points.
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According to Wikipedia:
Here's another source that concurs:
Early in Anthony Trollope's 1864 novel "Can you Forgive Her?" the heroine, Alice Vavasor, writes a letter that is central to the novel's plot development. She breaks off her (second) engagement by writing a letter, addressing her fiancee "Dear John" and then repeating this same greeting several times throughout her letter. So I imagine this very popular London novel was the origin of the concept, and that the idiom was further popularised in the USA during the second World War as suggested.