Rheological measurements are utilized to examine the yielding behavior of a polystyrene (PS) core and poly(N-isopropylacrylamide) (PNIPAM) shell microgel system with varying shell/core ratio. For a shell/core ratio of 0.15 at high concentrations, the suspensions show a typical hard sphere (HS) yielding response where the loss modulus (G '') exhibits a single peak due to cage breaking. As a result of tighter cages and less cage distortion prior to yielding, the peak location of G '' decreases with volume fraction. For a shell/core ratio of 1.10, which behaves like a soft jammed glass at high concentration, the suspensions exhibit a one-step yielding behavior similar to that of HS glass. However, the location of the peak in G '' increases with volume fraction, demonstrating the important role of particle deformation in the breakage of cages. For an intermediate shell/core ratio of 0.34, the system displays a two-step yielding behavior, as observed in previous reports for attractive glasses. By increasing the volume fraction, the strain of the first peak increases while the second one decreases. In addition, as the effective volume fraction increases to 112%, the two peaks merge into one broad peak. It is demonstrated that the first peak of G '' is due to deformation of the shell, and the second peak of G '' is attributed to cage breaking as a result of the cores colliding with each other. Combining these results, a yielding state diagram from typical HS to soft jammed glass is demonstrated.