I don’t claim to have the best ear for music, but it sounds to me like the backup singer here is juuuuust a fraction of a second off the lead vocalist. I’m going to conclude that this was done deliberately, because a) it works, and b) no one can claim that The Band was anything but a great … band.
Robbie Robertson called it “staggered vocal harmony”.
“We didn’t really get it that organised until we recorded it,” said Robertson. “And it came pretty simply. There was something about it that was so natural that it didn’t take a lot of organising… And then this thing at the end of the chorus where the voices stagger for that last line? It was a thing that came to me, like, in the last minute of working on the song. A lot of people say those spontaneous things are the best things and, well, this is one of those cases where that was very true!”
"I said, ‘Well, it’ll just be a back-up song in case some other things don’t work out’" – How Robbie Robertson wrote The Band's classic song, The Weight | MusicRadar
I’ve heard it on other songs, but suspect the effect was created in the mix rather than a conscious choice by the singers. However, there is a live recording of The Who performing “Listening to You” on The Kids are Alright soundtrack where Pete is staggering his vocals with Roger especially on the last word of each line.
Something like this?
Billy Bragg & Wilco - “The Unwelcome Guest”
(Jeff Tweedy’s background vocals don’t kick in until around the 1:50 mark, if you want to jump ahead.)