Non-orthogonal multiple access (NOMA) is considered a promising candidate for future mobile networks due to its ability to provide improved spectral-efficiency, massive connectivity and low latency. As such, studying the bit error rate (BER) performance of NOMA is crucial, particularly as its BER performance depends on the power assignment for each user. Therefore, this paper derives exact BER expressions under additive white Gaussian noise (AWGN) and Rayleigh fading channels for an arbitrary number of NOMA users, where each user employs quadrature amplitude modulation (QAM) with an arbitrary modulation order. Furthermore, the power coefficient bounds (PCB), which ensure fairness between users and solve the constellation points ambiguity problem, are derived for the two and three users NOMA system with arbitrary, but identical, modulation orders. However, the procedure to find these bounds for any modulation orders are exemplified. In addition, this paper finds the optimal power assignment that minimizes the system’s average BER for N=2 and 3 users cases. The integrity of the analytical expressions is verified by Monte Carlo simulations, where the results give a valuable insight on the system’s BER performance and power assignment granularity. It is shown that the feasible power assignment range becomes significantly small as the modulation order, or the number of users, increases, where the BER performance degrades due to the increased inter-user interference (IUI).