A fractal-looking basin boundary for forced periodic motions of a particle in a two-well potential is observed in numerical simulation. The fractal structure seems to be correlated with the appearance of homoclinic orbits in the Poincar\'e map as calculated by Holmes using the method of Melnikov. Below this critical forcing amplitude the basin boundary appears to be smooth and nonfractal. This example raises questions about predictability in nonchaotic dynamics of nonlinear systems.