The fundamental group of a closed hyperbolic 3-manifold is known to act geometrically on a CAT(0) cube complex. We ask whether the same is true for the fundamental group of negatively curved 3-pseudomanifolds, i.e., 3-manifolds with isolated singularities. While many 3-pseudomanifolds are cubulated, such as those arising from RACGs and strict hyperbolization, in this talk we give the first examples of closed 3-pseudomanifolds that are locally CAT(-1) but whose fundamental group cannot be cubulated. These examples are obtained from certain compact hyperbolic 3-manifolds with totally geodesic boundary by coning off the boundary components. This is joint work with J. Manning.