The impacts of heat waves in a warming climate depend not just on changing temperatures but also on changing humidity. Using 35 simulations from the Community Earth System Model Large Ensemble (CESM LENS), we investigate the long-term evolution of the joint distribution of summer relative humidity (RH) and daily maximum temperature () in four U.S. cities (New York City, Chicago, Phoenix, New Orleans) under the high-emissions Representative Concentration Pathway (RCP) 8.5.  We estimate the conditional quantiles of RH given by quantile regression models, using functions of temperature for each month and city for three time periods (1990-2005, 2026-2035, and 2071-2080). Quality of fit diagnostics indicate that these models accurately estimate conditional quantiles for each city. As expected, each quantile of increases from 1990-2005 to 2071-2080, while mean RH decreases modestly. For a fixed , the high quantiles of RH (and thus of heat index and dew point) increase from 1990-2005 to 2071-2080 in all four cities. This result suggests that the health impacts of a day of a given will increase in a warming climate due to the increase of RH. Conditional upon a fixed quantile of , the median and high quantiles of RH decrease, while those of heat index and dew point both increase. This result suggests that, despite a modest decrease in median relative humidity, heat stress impacts in a warming climate will increase faster than temperatures alone would indicate.