The orthogonal frequency division multiplexing (OFDM) and the non-orthogonal multiple access (NOMA) scheme are presented as promising techniques to meet the requirement of fifth-generation (5G) communication systems. Although much attention has recently been devoted to study these techniques, some scenarios have still been less explored. Considering that a fundamental part of any communication system is the use of power amplifiers, this paper presents an analytical evaluation of the bit error rate (BER) of NOMA-OFDM systems in the presence of a high power amplifier (HPA) with memory. Considering that the non-linear distortions generated by the HPA can be modeled using a polynomial model with memory, new theoretical expressions are developed to obtain the BER of the system. Specifically, exact BER expressions for a downlink NOMA-OFDM system with two users are presented and verified by Monte Carlo simulation results. The obtained numerical results demonstrate that the performance degradation of both users is highly dependent on the non-linear distortions, even when the successive interference cancellation (SIC) technique is performed perfectly.