In this paper, we investigate the feasible signal-to-interference-plus-noise-power-ratio (SINR) region of a multi-cell downlink non-orthogonal multiple access (NOMA) system with successive interference cancellation (SIC) technique. Based on the Perron-Frobenius Theory, we first derive a necessary and sufficient condition for an SINR vector to be feasible under a fixed SIC decoding order. The feasible SINR region of the system is then given by the union of the feasible SINR regions under all possible decoding orders. Next, we derive a necessary condition for an optimal SIC decoding order in the sense that it can achieve the SINR region boundary. Based on this condition, we further propose an efficient algorithm that can approximate the SINR region boundary. Simulation results are provided to validate the efficiency of the proposed algorithm and compare the performance of NOMA and orthogonal multiple access.