We report on ROSAT HRI observations of the nearby powerful radio galaxies 3C 33 and 3C 111, which both have detected optical hot spots. We find nuclear X-ray sources in both objects, but no X-ray emission from the hot spots. This confirms the presence of a high-energy cutoff in the spectrum of synchrotron-emitting electrons. Since these electrons necessarily scatter the synchrotron photons by the inverse-Compton process, our upper limits on the X-ray fluxes of the hot spots allow us to set lower limits of a few nanotesla on their magnetic flux density, close to or greater than the fields implied by equipartition of energy between radiating particles and magnetic field.