There’s a whole lot of “major release flop, low-budget straight-to-video/made for TV sequel” in this category, but there are still some standouts, i.e. 28 Days/Weeks Later and 48/Another 48 Hours stick out just on the first page.
I give Jean de Florette and Manon of the Spring credit. They were two good films that worked well together but was each able to stand up its own. And while they were critical and popular successes, they didn’t go to the well again (no pun intended) to try to make a third film.
I don’t know if this counts, but The Godfather and The Godfather Part II are generally considered among the greatest films ever made. Unfortunately, there’s also The Godfather Part III. This is not generally considered to be as good. Frances Ford Coppola has said that he thinks that The Godfather and The Godfather Part II should be consider the only two films in the series. He has said that The Godfather Part III should be considered just an epilogue to the series.
I have a personal weakness for American Graffiti/More American Graffiti. Great soundtracks, and the second movie gave a lot more depth to the epilog of the first. Richard Dreyfuss was too big to come back, but everybody else did.
It’s been years since I saw it, but I remember thinking “The Color of Money” was not great, but not bad. “The Hustler”, though, was a masterpiece. The greatness of the first spills over and fills in the deficiencies of the second. Cumulatively, its a great duology.
This only sorta counts, because the two movies aren’t formally related, just tonally similar, both written by Neil Simon, and with Peter Falk in similar roles:
The Mark of Zorro and Don Q Son of Zorro, Douglas Fairbanks’ two shots at the character. Also, Richard Lester’s The Three Musketeers and The Four Musketeers, though admittedly they adapted halves of the same novel. This was my favorite cast for these characters, Michael York and Oliver Reed et al.
The first duology was The Birth of a Nation and The Fall of a Nation. They don’t get my nomination, not only because I never saw the second part.
If you Google on the terms “Coppola”, “The Godfather III”, and “epilogue”, you’ll get a slew of references to how Coppola has said that the first two films should be considered a series and the third one just an epilogue. Some of those references says specifically that Coppola said this in the director’s commentary to the DVD’s of the films. I’ve never listened to that commentary, so I can’t attest to it. The references to what Coppola supposedly said claim that Coppola says that he and Puzo wanted to title the third film The Death of Michael Corleone, but Paramount refused to allow them to do so. Some references say that the TV re-editing of the first two films, usually referred to as The Godfather Saga or The Godfather: The Complete Novel for Television, which has some additions of scenes cut from those films, ought to be considered the definitive version of those two films and that The Godfather Part III is just a later afterthought.
And that’s the limit of my knowledge of the subject. There’s a lot of material published on those movies, little of which have I read. If you want to know more, you’ll have to do that reading yourself.