We present Chandra observations of a small sample of nearby classical double radio galaxies that have more than one radio hot spot in at least one of their lobes. The X-ray emission from the hot spots of these comparatively low-power objects is expected to be synchrotron in origin, and therefore to provide information about the locations of high-energy particle acceleration. In some models of the relationship between the jet and hot spot, the hot spots that are not the current jet termination points should be detached from the energy supply from the active nucleus, and therefore not capable of accelerating particles to high energies. We find that in fact some secondary hot spots are X-ray sources, and thus probably locations for high-energy particle acceleration after the initial jet termination shock. In detail, however, we show that the spatial structures seen in X-rays are not consistent with naive expectations from a simple shock model: the current locations of the acceleration of the highest energy observable particles in powerful radio galaxies need not be coincident with the peaks of radio or even optical emission.