Recent experiments show that the CoCrFeNiPd high-entropy alloy (HEA) is significantly stronger than CoCrFeNi and with nanoscale composition fluctuations beyond those expected for random alloys. These fluctuations were suggested to be responsible for strengthening. Here, a recent parameter-free theory for initial yield strength in fcc random alloys is shown to predict the strength of CoCrFeNiPd in good agreement with experiments. The strengthening is due mainly to the large misfit volume of Pd in CoCrFeNi, indicating that the effects of the non-random composition fluctuations are secondary. Analyses of strength variations and strengthening-associated length scales help rationalize why dislocation motion is insensitive to such fluctuations. These findings point to the value of theory for understanding the mechanical behavior of HEAs.