The coarse Baum–Connes conjecture is a foundational statement connecting coarse K-homology with K-theory of Roe algebras. It has long been known to be false for certain spaces constructed as infinite disjoint unions of expander-like objects. The goal of this talk is to present recent joint work with Kitsios and Schick, in which we prove a conjecture of Roe by showing that even fairly tame Riemannian manifolds (specifically, certain warped cones) violate the coarse Baum–Connes conjecture. In both cases the failure of Baum-Connes relies on some spectral gap type properties of the underlying metric spaces.