The mechanical properties due to solid solution strengthening are explored within the single phase fcc domain of the Co-Cr-Fe-Mn-Ni high entropy alloy (HEA) system. This is achieved by combining an efficient and reproducible metallurgical processing of alloys to X-ray diffraction and nanoindentation characterization techniques, thus enabling to get access to 24 different bulk alloys. Large variations of nanohardness are seen with composition. Experimental results are rationalized in terms of lattice misfit and elastic constant variations with alloy-composition, through the use of an analytical mechanistic theory for the temperature-, composition-and strain-rate-dependence of the initial yield strength of fcc HEAs, with predictions made using only experimental inputs. The good agreement obtained by comparing model predictions to experiments provides the basic framework for mechanical properties optimization within the Co-Cr-Fe-Mn-Ni system; the approach could be systematically applied to all classes of fcc HEAs.