This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments.