What movies and TV shows have mail men/postal clerks/etc in them?
There was the Night Court episode with the clerk who was a really really slow talker: Night Court - S4E15 Tuttle Clip - YouTube
And MIB II had all the people in the post office be aliens.
Also, Cliff Claven from Cheers.
What else?
The Postman, although the character is Italian, not USPS.
Not to be confused with The Postman released three years later, with Kevin Costner as the last person with any semblance of a link to the former government in a post-apocalyptic America.
The postal employees in Miracle on 34th Street end up being vital to the plot.
Hello? Neuman.
Also Mike Wobbles and David McFeely.
George Wilson from Dennis the Menace is a retired mailman. (Okay, it’s primarily a comic strip but it’s been adapted for television and movies.)
Jacobs Ladder
Jack Albertson (uncredited, but essential) in Miracle on 34th Street
Three Days of the Condor shows the darker side of Mr Zip.
Newman (first name unknown).
My favorite Newman quote: “I’ll tell you a little secret about ZIP codes: they’re meaningless.”
Who could forget Mr. McFeely, from Mr. Rogers, perhaps the most inappropriate name for a children’s show character in television history.
For about the last 20 years, none seem to technically be American letter carriers. USPS, like Amtrak and Scouting USA, seems to be fiercely protective of its trademarks, so the uniforms always have to be knockoffs or incomplete.
Does an Adam Sandler film count?
“I now pronounce you Chuck and Larry”
Extreme nitpick. Oooooh this is tricky - no he isn’t a postman.
Because:
Whilst a lengthy chunk of Jacob’s life is portrayed, in which he is a postman, this life does not in fact occur. (The film uses the same device as Occurrence at Owl Creek Bridge by Ambrose Bierce.
Remember the time Hannibal Lecter filled in for him? His version of “Speedy Delivery” involved fava beans and a nice Chianti.
Here is the same actor playing a mailman in How I Got Into College, by the same director as Better Off Dead.
OP: What a completely bizarre question.
“McFeely” was Fred Rogers’ middle name and the last name of his maternal grandfather.
Came to post this. He’s the one who decided to forward all the “Dear Santa” letters to Kris Kringle in the courtroom. Great scene when they bring in dozens of bags of mail and pile them on the judge’s bench! 
This is the original version – haven’t watched any of the others.