My impression is that Stanley's *partionability conjecture* and his *depth conjecture* (which was shown to imply the partionability conjecture in 2008) were both believed to be true, until Duval, Goeckner, Klivans, and Martin found a counterexample to the partionability conjecture in 2015. See this AMS survey article: https://www.ams.org/journals/notices/201702/rnoti-p117.pdf. This fits with a few other examples already mentioned (the Hauptvermutung, the Hirsch conjecture, ...) which warn us that although simplicial complexes/polytopes may appear to be intuitively simple objects, they can in fact be extremely complicated.