We report significant hard X-ray excesses in the energy range 2-8 keV for two nearby isolated neutron stars: RX J1856.6-3754 and RX J0420.0-5022. These neutron stars have previously been observed in soft X-rays to have nearly thermal spectra at temperatures ∼100 eV, which are thought to arise from the warm neutron star surfaces. We find nontrivial hard X-ray spectra well above the thermal surface predictions with archival data from the XMM-Newton and Chandra X-ray telescopes. We analyze possible systematic effects that could generate such spurious signals, such as nearby X-ray point sources and pileup of soft X-rays, but we find that the hard X-ray excesses are robust to these systematics to the extent that is possible to test. We also investigate possible sources of hard X-ray emission from the neutron stars and find no satisfactory explanation with known mechanisms, suggesting that a novel source of X-ray emission is at play. We do not find high-significance hard X-ray excesses from the other five Magnificent Seven isolated neutron stars.