We discuss possible competition between magnetic and quadrupole Kondo effects in non-Kramers doublet systems under cubic symmetry. The quadrupole Kondo effect leads to non-Fermi-liquid (NFL) ground state, while the magnetic one favors ordinary Fermi liquid (FL). In terms of the j-j coupling scheme, we emphasize that the orbital fluctuation must develop in the vicinity of the NFL-FL boundary. We demonstrate a change of behavior in the f-electron entropy by the Wilson's numerical

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... roup (NRG) method on the basis of the extended two-channel Kondo exchange model. We present implications to extensively investigated PrT_2X_20 (T=Ti, V, Ir; X=Al, Zn) systems that exhibit both quadrupole ordering and peculiar superconductivity. We also discuss the magnetic-field effect which lifts weakly the non-Kramers degeneracy. Our model also represents the FL state accompanied by a free magnetic spin as a consequence of stronger competition between the magnetic and the quadrupole Kondo effects.