What happened to the post office attempting to deliver mail?

Admittedly, it was in the late 40’s an early 50’s me & my siblings were required to write to our Grandma.
The address used was:

Grandma
Town, Tx

No zip codes in those day of course.
She got the letters.

I think they also got the clue from the return address who’s Grandma it was supposed to go to.

Also, of all the Grandmothers in that town, she was “The Grandma.”