This letter examines the bit error rate (BER) performance of downlink non-orthogonal multiple access networks for binary phase-shift keying modulation. Exact BER expression is derived for each user in closed-form under additive white Gaussian noise and Rayleigh fading channels in perfect and imperfect successive interference cancellation (SIC) cases. Next, in perfect SIC case, the asymptotic BER expression in a high signal-to-noise ratio (SNR) region is obtained to express the behavior of the network with diversity and array gains. On the other hand, in imperfect SIC case, the upper bound for BER is attained, and at high SNR values, the BER reveals an error floor, and hence a zero diversity gain is achieved. Then, a feasible range of power allocation coefficients is found such that a good BER performance can be provided for each user. Finally, through simulations and software-defined radio-based real-time tests, analytical results are validated.