Many centralized matching markets are preceded by interviews between participants. We study the impact on the final match of an increase in the number of interviews for one side of the market. Our motivation is the match between residents and hospitals where, due to the COVID-19 pandemic, interviews for the 2020-21 season of the National Residency Matching Program were switched to a virtual format. This drastically reduced the cost to applicants of accepting interview invitations. However, the reduction in cost was not symmetric since applicants, not programs, previously bore most of the costs of in-person interviews. We show that if doctors can accept more interviews, but the hospitals do not increase the number of interviews they offer, then no previously matched doctor is better off and many are potentially harmed. This adverse consequence is the result of what we call interview hoarding. We prove this analytically and characterize optimal mitigation strategies for special cases. We use simulations to extend these insights to more general settings.