Tunneling and theoretical studies have suggested that Andreev bound states form at certain surfaces of unconventional superconductors. Through studies of the temperature and field dependence of the in-plane magnetic penetration depth lambda(ab) at low temperature, we have found strong evidence for the presence of these states in clean single crystal YBCO and BSCCO. Crystals cut to expose (110) surfaces show a strong upturn in lambda(ab) at around 7 K, when the field is oriented along the c axis. In YBCO this upturn is completely suppressed by a field of approximately 0.1 T.