In this letter, we investigate the performance of non-orthogonal multiple access (NOMA), under the assumption of generalized Gaussian noise (GGN), over Rayleigh fading channels. Specifically, we consider a NOMA system with $L$ users, each of which is equipped with a single antenna, and derive an exact expression for the pairwise error probability (PEP). The derived PEP expression is subsequently utilized to derive a union bound on the bit error rate (BER) and to quantify the diversity orders realized by NOMA users in the presence of additive white (AW) GGN. Capitalizing on the derived PEP expression and the union bound, the error rate performance of NOMA users is further evaluated for different special cases of AWGGN. The derived analytical results, corroborated by simulation results, show that the shaping parameter of the GGN $(α)$ has negligible effect on the diversity gains of NOMA users, particularly for large $α$ values. Accordingly, as in the case of additive white Gaussian noise (AWGN), the maximum achievable diversity order is determined by the user's order.