The third song to reach #1 by two different artists was ‘Please Mr. Postman,’ though arguably the best-known version of the track belongs to a band that didn’t send it up the charts.
‘Please Mr. Postman’ was the 1961 debut single of The Marvelettes, and the first song on the Motown label to reach #1. The all-female quartet was one of Motown’s biggest successes before they were eclipsed by The Supremes a few years later.
This song received its next high-profile release in 1963, when The Beatles recorded it for their second album, With the Beatles. Their version is the one I know best, and I’ll go out on a limb and say it’s even better than the original. The recording was released as the B-side to ‘Roll Over Beethoven’ in some countries and didn’t come close to hitting #1 in the U.S.
In 1975, however, the Carpenters included the song on their album Horizon, and scored the third (and final) Billboard #1 hit of their career. Their recording doesn’t differ much from the original, and sounds like it could easily have been released in the 50s or 60s rather than halfway through the 70s.
(Wait) Wait, Mister Postman
(Please, mister Postman, look and see) Oh yeaheaheah
(If there’s a letter in your bag for me) Please, please, mister Postman
(Why’s it takin’ such a long time?) Oh yeah
(For me to hear from that boy of mine)
There must be some word today
From my boyfriend, so far away
Please, mister Postman, look and see
If there’s a letter, a letter for me
I’ve been standin’ here waitin’, mister Postman
So patiently
For just a card or just a letter
Sayin’ he’s returning’ home to me
Mister Postman (Mister Postman, look and see), oh yeaheah
(If there’s a letter in your bag for me) Please, please, mister Postman
(Why’s it takin’ such a long time?) Oh yeah
(For me to hear from that boy of mine)
So many days you passed me by
See the tears standin’ in my eyes
You didn’t stop to make me feel better
By leavin’ me a card or a letter
Mister Postman (Mister Postman, look and see), oh yeah
(If there’s a letter in your bag for me) Please, please, mister Postman
(Why’s it takin’ such a long time?)
Why don’t you check it and see one more time for me?
You gotta wait a minute, wait a minute
Wait a minute, wait a minute
Ooohooohooo, mister Postman (Mister Postman, look and see)
Come on, deliver the letter, the sooner, the better
Meet Me In Montauk 
I’m most familiar with the original version of this song, but tend to agree that I prefer the Beatles take on it.
Meanwhile, I’m curious if Disney paid the Carpenters decent money for that video advertisement of Disneyland.