It is shown, by giving an explicit example, how the conditions imposed on physically admissible states can affect the pattern of supersymmetry breaking and, as a consequence, the values of physically interesting quantities such as the Witten index. As a by-product an unusual kind of explicit supersymmetry breaking is observed, where the supersymmetry algebra still holds formally, as well as a spontaneous breakdown of supersymmetry, which is not induced by instantons.