If we invent a subcategory of films made before 1960 that you actually saw before 1960 (sorry, young folks can’t play): Old Yeller and Treasure Island, both of which affected me profoundly. If I had seen Casablanca or Kane back then, I would have been very bored!
The greatest movie before 1960 is Citizen Kane. For the total brilliance of Orson Wells. See how movies looked before CK and see how they looked after. When put in context it’s the greatest movie of all time. However, after seeing so many other movies which came after, all of which owe a debt to this great movie, CK is a bit slow moving. Without the big reveal and the immediacy of William Randolph Hurst’s real world influence it loses its impact. Therefore…
The best movie pre-1960 is Casablanca.
You are right Auntie Pam…thanks for correcting me. 
Movies need to broken down into different categories to answer this question. Casablanca is a great all around movie but Oz stands out above all others. Oz can’t really compete as a storyline but Casablanca can’t compete visually. Oz rises about everything else in special effects and cinematography. The creative costumes and sets are just over the top for the time period and the actors meld into it flawlessly. I’m frankly amazed the movie was ever made considering it was essentially a kids movie.
Before the days of cable TV this was THE movie of the year to watch on TV. I can still remember seeing it for the first time in color.
Seven Samurai - so great it gets constantly remade as good films (Magnificent Seven, Bug’s Life…Oh, and the 3 Amigos).
Casablanca got remade as Barb Wire.
Forty-nine posts before someone mentioned Grand Illusion, and only as an also-ran at that.
Citizen Kane isn’t even the best Orson Welles movie (note the second “e”) before 1960. Yes, it was groundbreaking and bears repeated viewing and is a wonderful all around movie. If you liked CK, you will love Touch of Evil and The Lady From Shanghai. Also see Othello and The Stranger.
That’s why I made a distinction between greatest and best.
I wrote the OP, and I’m happy for people to use their own criteria. Somebody else can start the “What makes a movie great?” thread.
Darn good question. I have no idea. Is it a compelling story? The acting? The set design and costumes? The ability to relate to the characters? I think we each have our own criteria. To use modern examples, Spielberg’s Lincoln and The King’s Speech: Lincoln is the greater movie, The King’s Speech is the better movie.
I don’t think it could be resolved, but maybe it deserves a thread as Skald suggests. I think of things like acting, cinematography, plot, characters, and special stuff like effects or music or dance. But then I use Bridge On the River Kwai as an example why this doesn’t work, lots of great acting, plot, characters, the bridge blowing up with the train crossing it, and the catchy tune. But it’s still a flawed movie as a whole to me. I tend to go with something like Oz because it has all the individual criteria and that magic quality of a great movie on top of all that.
One thing I wouldn’t count are minor continuity problems.
Sunrise (1927)
Some people will find its story implausible or be horrified at a plot point that I won’t spoil. But the film is so shockingly beautiful in every facet–from the cinematography to art design to performances–that it likely won’t ever be displaced at the top of my own list of masterpieces.
It can be enjoyed as a narration of literal events or as a metaphor for the course of a whole marriage. Its characters can be seen both as people and as personifications. It opens profound questions about the meaning and limits of love and forgiveness. If you haven’t seen it before, the final scenes will smack you with disbelief at the cosmic justice (or injustice) on display. And all while the Movietone score accompanies it with “Les préludes” and “Oh Promise Me”.
You might have something there. It’s the only full length silent movie I’ve ever been able to sit through, and thanks to TCM, I’ve tried many. Sunrise was mesmerizing.
I shall have to watch this. Sounds interesting.