If anyone is interested, I think I figured it out.
Some of the media files referenced in the project were viewed with their file extensions in the filenames. So when the file “video.mov” was copied by Media Manager multiple times, each duplicate was numbered as follows: “video.mov1”, “video.mov2” etc. Numbering the files at the end of the file extension changed the file type altogether, making those files unreadable by final cut.