This paper investigates the average bit error rate (BER) of a reconfigurable intelligent surface (RIS)-assisted downlink non-orthogonal multiple access (NOMA) system underN agamani-m fading. The base station communicates binary phase-shift keying (BPSK) data with the near and far users through an RIS unit enabled with discrete phase shifter elements. The total number of RIS elements is divided into two parts, each supporting an individual user. We use generalized-K and mixture gamma-based approximation to present an accurate analysis for an arbitrary number of RIS elements. We provide upper and lower bounds on the average BER for uniform quantized phase error. The bounds are applicable for 2-bit and higher level quantization. Both bounds render a better approximation with an increase in the quantization level. Moreover, the lower bound is tighter than the upper bound and, specifically, for 3-bit and higher quantization levels, offers an accurate approximation with the simulation results. We also present the asymptotic average BER to simplify the analysis at high signal-to-noise ratios. Furthermore, we demonstrate some numerical results that may help design similar systems.