That phrasing will always make the hear think, "the day after what?"
If the event has been mentioned in an earlier sentence and is understood, then there is no problem. But in a standalone sentence as in your original examples it would be unclear.
It is also unclear in the movie title, but that is an allusion, a clever and intentional device to draw attention to and create interest in the movie. As an example of artistic license or of clever marketing, it is outstanding. As an example of unambiguous English it would be a complete failure. Nobody could tell you what day "The Day After" refers to without knowing something about the movie.
I'd also say it's possible that the use of 'The Day After" in the movie title was referencing the idea that with such a world changing event a phrase which would normally require a specificity would no longer need one since it would become contextually obvious to everyone who lived through the event.
In much the same way that if one says "my friend was there on nine eleven" to most US citizens they assume it is the date of the disaster and the location was New York. Whereas if you said "my friend was there on four sixteen" people would not have an immediate contextual understanding.
(c) I bought the car last Tuesday and the day after it broke down.
This could also be seen as a sentence fragment - an incomplete sentence. Its as if the entire sentence should be:
"I bought the car last Tuesday and the day after it broke down some surprising thing happened".
Of course most people will infer from the context that you mean the day after you bought it, but it is an ambiguity that should be eliminated by constructing a better sentence.
I would prefer; I bought the car last Tuesday, and the next day it broke down.
But this doesn't help you in your quest for a sentence that uses "the day after", but I think that phrase will almost always be an incomplete fragment, and, as Mike says (and as every US high school English teacher will say) it makes me think... "the day after what?"