Observations of red-sequence galaxies in both the local and high-redshift Universe and hierarchical galaxy formation models make firm predictions for the stellar populations of elliptical and S0 galaxies (early-type galaxies or ETGs). Three of the most robust are (1) ETGs of a given stellar mass form stars earlier and thus should have older stellar population ages in the highest-density environments than those in lower-density environments; (2) the most-massive ETGs in the densest environments should have a small spread in stellar population ages, and (3) in all environments, smaller ETGs should have younger stars (`downsizing'). To test these predictions, I present stellar population parameters of twelve elliptical and S0 galaxies in the Coma Cluster around and including the cD galaxy NGC 4874, based on spectra obtained using the Low Resolution Imaging Spectrograph on the Keck II Telescope. I extend the sample with literature data to study trends with velocity dispersion and mass. I show that ETGs in the Coma Cluster violate predictions (1) (by comparison with field ETGs) and (3), but apparently red-sequence galaxies do not. These results may raise interesting problems with current models of galaxy formation.