We present recent results of quiescent X-ray observations of recurrent novae (RNe) and related objects. Several RNe are luminous hard X-ray sources in quiescence, consistent with accretion onto a near Chandrasekhar mass white dwarf. Detection of similar hard X-ray emissions in old novae and other cataclysmic variables may lead to identification of additional RNe candidates. On the other hand, other RNe are found to be comparatively hard X-ray faint. We present several scenarios that may explain this dichotomy, which should be explored further.