In 1908 Steinitz and Tietze formulated the Hauptvermutung ("principal conjecture"), according to which, given two triangulations of a simplicial complex, there exists a triangulation which is a common refinement of both.
This was important because it would imply that the homology groups of a complex could be defined intrinsically, independently of the triangulations which were used to calculate them.
Homology is indeed intrinsic but this was proved in 1915 by Alexander, without using the Hauptvermutung, by simplicial methods.
Finally, 53 years later, in 1961 John Milnor (some topology guy, apparently) proved that the Hauptvermutung is false for simplicial complexes of dimension $\geq 6$.
