Visible light communication (VLC) systems are expected to provide remarkably high speed indoor communications and effective ubiquitous connectivity. However, the key limitation of such systems is the narrow modulation bandwidth of the light sources. Based on this, non-orthogonal multiple access (NOMA) has been recently proposed as an effective method that can enhance considerably the spectral efficiency of indoor downlink VLC systems. In this context, the present work is devoted to the evaluation of the bit-error-rate (BER) performance of NOMA-based VLC systems. Specifically, a novel closed-form expression is first derived for the BER of the considered set up, by also taking into account the realistically incurred cancellation errors and interference terms. The validity of the derived expressions is verified through extensive comparisons with respective results from Monte Carlo simulations, while their algebraic representation is relatively simple, which renders them convenient to handle both analytically and numerically. This leads to meaningful insights on the behavior and performance gains achieved, thanks to the adoption of NOMA, which are particularly useful in future design and deployment of VLC systems.