"Fat" meant something like "phat." "Fat" meaning "overweight" came soon after.
15 March 1956, Pasadena (CA) <Ä±>Independent>/Ä±>, pg. A2:
You know you're getting old when the kids look down their collective noses at you for saying "Crazy" when things are alright. Teenager buddy clued the Museum the other day, "Man, crazy is out of it -- you gotta say 'fat city' now." Oh well.
13 April 1961, Washington Post, pg. A1:
"Wow, this is Easy Street in a fat city," said a white Peking Duckling, lying back maharajah-style.
6 June 1963, Oakland (CA) Tribune, pg. 30:
The winner of the 1963 Joseph Henry Jackson Award of $1,000 is Leonard Gardner, a 29 year old San Francisco writer. Gardner won the award, named for the late critic and author, for a novel in progress called "Fat City."