A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller. It is shown in this paper that a generic $C^1$ vector field on a closed $n$ -manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open-dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of $C^1$ vector fields on a 5-manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.