The aim of this paper is to reveal the importance of hybrid successive interference cancellation (SIC) and power adaptation (PA) for improving transmission robustness of uplink non-orthogonal multiple access (NOMA). Particularly, a cognitive radio inspired uplink NOMA communication scenario is considered, where one primary user is allocated one dedicated resource block, while M secondary users compete with each other to be opportunistically served by using the same resource block of the primary user. Two novel schemes are proposed for the considered scenario, namely hybrid SIC with PA (HSIC-PA) scheme and fixed SIC with PA (FSIC-PA) scheme. Both schemes can ensure that the secondary users are served without degrading the transmission reliability of the primary user compared to conventional orthogonal multiple access (OMA) based schemes. Rigorous analytical results are presented to evaluate the performance of the proposed two schemes. It is shown that both schemes can avoid outage probability error floors without any constraints on users' target rates in the high SNR regime. Furthermore, it is shown that the diversity gain achieved by the HSIC-PA scheme is M, while that of the FISC-PA scheme is only 1. Numerical results are provided to verify the developed analytical results and also demonstrate the superior performance achieved by the proposed schemes by comparing with the existing HSIC without PA (HSIC-NPA) scheme. The presented simulation results also show that HSIC-PA scheme performs the best among the three schemes, which indicates the importance of the combination of HSIC and PA for improving transmission robustness.