We study the approximation in Lebesgue spaces of wave- let bi-frame systems given by translations and dilations of a finite set of generators. A complete characterization of the approximation spaces associated with best m-term approximation of wavelet bi-framelet systems is given. The characterization depends, in general, on the number of vanishing moments of the generators. However, in some cases it is possible to get rid of this restriction, but at the price of replacing the canonical expansion by another linear expansion. This is done by expanding a suitable wavelet in the wavelet bi-frame system.