We discuss a new methodology for statistical Blind Source Separation (BSS) in a change-point regression setting. More precisely, one observes a linear mixtures of piecewise constant signals (sources) taking values in a known finite alphabet. The aim in this models is to identify the unknown mixing weights and sources, including the number of sources, from noisy observations of the mixture. This BSS problem occurs, for example, in digital communications and cancer genetics. Under weak identifiability conditions we obtain exact recovery within a neighborhood of the mixture. Based on this we provide uniformly honest lower confidence bounds and estimators with exponential convergence rates for the number of source components. With this at hand, we obtain consistent estimators with almost optimal convergence rates and asymptotically uniform honest confidence statements for the weights and the sources. We explore our procedure with a data example from cancer genetics, where one aims to assign copy-number variations from genetic sequencing data to different tumor-clones and their corresponding proportions in the tumor.