I'm not sure about that. It would be true with a spherical cow, but I don't think it's true when real people are involved because people sweat to cool down. I believe that phase change of water evaporating off your skin should continue to cool you down no matter how hot it gets, like a swamp cooler in Arizona. A fan should help this by rapidly replacing the humid air around you with more dry air.
(However my understanding is that evaporation cooling stops working once humidity hits 100%)