The Fixed Anvil Temperature (FAT) hypothesis proposes that upper‐tropospheric cloud fraction peaks at a special isotherm that is independent of surface temperature. It has been argued that a FAT should result from simple ingredients: Clausius‐Clapeyron, longwave emission from water vapor, and tropospheric energy and mass balance. Here, the first cloud‐resolving simulations of radiative‐convective equilibrium (RCE) designed to contain only these basic ingredients are presented. This setup does not produce a FAT: the anvil temperature varies by about 40% of the surface temperature range. However, the tropopause temperature varies by only 4% of the surface temperature range, which supports the existence of a Fixed Tropopause Temperature (FiTT). In full‐complexity RCE simulations, the spread in anvil temperature is smaller by about a factor of two, but the tropopause temperature remains more invariant than the anvil temperature by an order of magnitude. In other words, our simulations have a FiTT, not a FAT.