An aperiodic and especially a chaotic attractor may consist of some number K >= 1 of bands (also denoted as connected components). Multi-band chaotic attractors (MBCAs) defined by the bandcount K > 1, represent a well-known phenomenon on the field of nonlinear dynamics and are often involved in several bifurcations. It is for instance well-known, that the period doubling cascade is typically followed by an inverse band merging cascade, which represents a sequence of MBCAs with p0 2^n bands, whereby n decreases from infinity to zero. We report a novel bifurcation scenario (bandcount adding scenario), which involves an infinite number of MBCAs organized not sequentially, but according to an infinite adding scheme. This adding scheme which is similar to the well-known Farey-trees, implies that between two MBCAs with bandcounts K1 and K2 there is an MBCA with K1 + K2 - K0 bands (with some constant offset K0). This scenario continues ad infinitum and resembles the period adding scenario known from many applications, but in contrast to this is formed by chaotic and not by periodic attractors. We study this scenario using a discontinuous map, which is actually considered by many authors as some kind of normal form of the discrete-time representation of many non-smooth systems of practical interest in the neighborhood of the point of discontinuity. By investigation of the structure of the 2D parameter space, we find out, that the bandcount adding scenario is related with discontinuityinduced codimension-1 bifurcations of unstable periodic orbits and with some specific discontinuity-induced codimension-2 bifurcations. Consequently, we describe the self-similarity of the chaotic area in parameter space and show, that in this area there are much more non-robust chaotic attractors than typically assumed.