A remarkable 1960 result of J. Glimm provides a sharp dichotomy for the representation theory of separable C*-algebras. One of its consequences is that a separable C*-algebra either has a unique (up to the unitary equivalence) irreducible representation or continuum many inequivalent irreducible representations. Using some ideas of Akemann and Weaver, I’ll prove that this conclusion is consistently false for nonseparable C*-algebras. Many open problems remain.