My parents had this problem when I was growing up. We lived in town A but due to being on the border of town B and in the middle of nowhere we got our mail from town B. Anything addressed to town A wouldn't be delivered. They ended up getting a PO box in town A and post office would put all the undeliverable stuff in it.

With a growing population town A decided to start delivering their mail something like 20-25 years ago. I can see it being a bigger problem now with autofill addresses and such.