They also did one called Trail Mix, but I forget what movie that ran with.
I recall two:
and
“No, I wear an extra pair of underwear.”
I believe I saw a showing of “The Critic” just before the ran Blazing Saddles in the theater in 1974. But they were unusual at that point.
I do remember seeing them routinely in the early 60s.
I remember them in the early/mid 70’s. But I was a kid, going to kids’ movies - it may have been more common to run a cartoon before ‘The Apple Dumpling Gang’ than ‘The Sting’ (which I also saw in the theater).
If you count cartoon shorts *attached *to movies, I saw a great new Road Runner/Coyote cartoon before a horrible movie about dog spies, just a few months ago.
Mid '90s. 
At the University of Wisconsin-Madison they showed a free movie on Fridays in a big lecture hall on campus and it almost always had a Looney Tunes short first.
I should have known someone would respond with this before I had a chance!
I saw “Dr. Strangelove” at the Paramount about, oh, 6 or 7 years ago, now, and they had a cartoon and a Three Stooges short before the feature presentation.
I saw an early Pixar short, Knick Knack, in front of Mel Gibson’s Hamlet, which would have been in 1990.
The last time I can remember seeing a cartoon in front of a regular feature movie before that would have been at a drive-in movie, probably in 1974 (though specifics on both the cartoon and the feature now escape me).
When I was in college c. 1980, I attended a Woody Allen film festival (Bananas and Sleeper were playing that particular night) at NARO Expanded Cinema in Norfolk, Virginia. A Three Stooges short was screened before the features.
I also saw several cartoons – including such relartive unknowns as James Hound (James Bond ripoff/tribute) and Robin Goodhood, featuring Roland Hood stealing from evil tax collector Rattfink – at drive-ins in the late '60’s and early '70’s. These theaters also sometimes screened documntary shorts – I remember one about the Moro people of the Philippines, and one with Ed Ames singing a song with “You Don’t Believe It’s True” repeated in the lyrics (I think this was an Ohio travelogue, which would make sense as I grew up in suburban Cleveland).
Yes, got music videos in NZ too in the early to mid-80s, the two I remember seeing more than once each were both Dire Straits videos: Money For Nothing, and Romeo and Juliet. And as GuanoLad said, we still used to get the occasional short up until about then too.
All the theaters had cartoons in the early 1960’s, unless, rarely, it was a serious movie like Dr Zhivago. The cartoons were generally shown between the two double features.
Back then, you also could come and go as you please, you could stay in the theater all day and see the movie over and over if you wanted to. I usually went to a show near the end of the previous double feature, to see if the second show was worth seeing.
The last time I distinctly remember seeing a cartoon before a movie was a Woody Woodpecker when my mom took me to Smokey and the Bandit (1977). It was at a miniplex in NE Indiana.
Aside from Pixar movies, I believe I saw “Duck Amuck” playing before a movie at Place Riel Theatre at the University of Saskatchewan. This would have been around 1992.
When I was a kid in the 50s. Cartoons and Newsreels.
Dreamcatcher, the 2003 Stephen King movie, had The Final Flight of the Osiris from The Animatrix playing in front of it.
I never watched the movie, but I saw the short a bunch of times while working in the movie theater. I had to tell a lot of people to stick around, because their feature would be starting in a few minutes, and this was just a short. It confused many many people.
I saw a cartoon in a downtown Chicago theater before a showing of either Death in Venice (1971) or The Night Porter (1974). It could’ve been a Pink Panther short, “Thank You, Masked Man” (1971 animation of a Lenny Bruce routine) or something like that. No cartoon before Manhattan (1979).
Probably “Toy Story.”
I remember that a cartoon was shown before “Who Framed Roger Rabbit,” but to answer what the OP is getting at, it seems the '70s were the point when theaters stopped showing cartoons on a regular basis, from my experience.
1996, movie was How the West Was Won in Cinerama.
As with several others, the “Crunch Bird” before “Sleuth” in 1972.
Around the same time I remember seeing a short where a young man shows up at the house of an elderly childless couple, says he’s their son, works his way into their hearts and then leaves. Can’t remember the title or the main film.
I remember seeing cartoons before movies a couple of times in the late 80s or early 90s. I think the theater was on some kind of nostalgia kick. They didn’t do it for very long.