The film also has a number of scenes that end abruptly, suggesting that it was probably cut down some before its release. If MGM cut it down, I can't blame them, since it runs over two hours as it is, but it might explain why the story (such as it is) is hard to follow.
In some ways I think the coming of CinemaScope wrecked Vincente Minnelli's career. Starting in 1954 all his movies were made in the 'Scope format (except for his remarkably bad last film, A Matter of Time). Minnelli was always a director who liked long takes; he said he was influenced by Max Ophuls, who believed that it was more interesting for the camera to move from one composition to another within a shot instead of cutting between compositions. Minnelli's pre-1954 movies are famous for their long takes, dazzling camera moves and heightened emotion, whether in musicals or melodramas. But when widescreen came in, camera movements became more cumbersome for a while (not so much because of the camera itself, as because it took a while for directors and cinematographers to figure out how to move between one 'Scope composition and another), and there was a pervasive idea that 'Scope movies should be filmed in long takes with minimal cutting and minimal camera movement. This wasn't just out of convenience; if you look at film-theory essays from the '50s, there was a strongly-held belief that it was aesthetically right to shoot a 'Scope film this way, that the format demanded that viewers be allowed to decide for themselves what they wanted to look at.
The result is that most of Minnelli's post-1954 movies have some great long takes and beautiful compositions, but a lot of static scenes where characters sit talking in the centre of the frame, with maybe one cumbersome camera movement at the very end of the shot. With some exceptions (like some of the big scenes from Some Came Running) his movies don't have a lot of energy or tension; they're slack and slow. This may be a partial explanation for why The Bad and the Beautiful, which he'd done with Houseman only three years before, feels so fast and brilliant and the melodrama of The Cobweb is kind of boring by comparison, despite the great cast. (The other part of the explanation is that The Bad and the Beautiful had a script that didn't suck.) His post-1954 musicals, including Gigi, are also a little low on energy.
The Cobweb is also known for having the first Hollywood score to make substantial use of Schoenberg's twelve-tone system; Leonard Rosenman, who'd studied with Schoenberg, was the hot young composer in Hollywood for 1955 only (when he did the two James Dean vehicles and this picture). Houseman hired him over the objections of MGM's music department and gave him free rein to write any kind of score he wanted, so he decided to write music that would elevate the story by portraying what was going on in the characters' minds -- which is to say, they're nuts. He unwittingly helped create the longstanding Hollywood rule where atonality is associated with madness; Jerry Goldsmith later used some twelve-tone procedures when he scored a movie about Sigmund Freud. I don't know that Rosenman's music actually helps the movie very much, not because of the Schoenbergian techniques but just because the actual material isn't very interesting and seems to call attention to itself rather than the movie. It certainly doesn't compare to David Raksin's score for The Bad and the Beautiful or Minnelli/Houseman's later over-the-top melodrama, Two Weeks in Another Town. To be honest, I've never heard a score by Rosenman that I really liked.