Via numerical simulation and experimental assessment, this study examines the use of origami folding to develop robotic jumping mechanisms with tailored nonlinear stiffness to improve dynamic performance. Specifically, we use Tachi-Miura Polyhedron (TMP) bellow origami -- which exhibits a nonlinear "strain-softening" force-displacement curve -- as a jumping robotic skeleton with embedded energy storage. TMP's nonlinear stiffness allows it to store more energy than a linear spring and offers improved jumping height and airtime. Moreover, the nonlinearity can be tailored by directly changing the underlying TMP crease geometry. A critical challenge is to minimize the TMP's hysteresis and energy loss during its compression stage right before jumping. So we used the plastically annealed lamina emergent origami (PALEO) concept to modify the TMP creases. PALEO increases the folding limit before plastic deformation occurs, thus improving the overall strain energy retention. Jumping experiments confirmed that a nonlinear TMP mechanism achieved roughly 9% improvement in air time and a 13% improvement in jumping height compared to a "control" TMP sample with a relatively linear stiffness. This study's results validate the advantages of using origami in robotic jumping mechanisms and demonstrate the benefits of utilizing nonlinear spring elements for improving jumping performance. Therefore, they could foster a new family of energetically efficient jumping mechanisms with optimized performance in the future.