We report that the partial substitution of Cu for Co has a significant impact on the magnetic properties of the Heusler-phase Weyl fermion candidate ZrCo2Sn. Polycrystalline samples of ZrCo2-x Cu x Sn (x = 0.0-1.0) exhibited a linearly decreasing ferromagnetic transition temperature and similarly decreasing saturated magnetic moment on increasing Cu substitution x. Materials with Cu contents near x = 1 and several other quaternary materials synthesized at the same x (ZrCoT'Sn (T' = Rh, Pd, Ni)) display what appears to be non-ferromagnetic magnetization behavior with spin glass characteristics. Electronic structure calculations suggest that the half-metallic nature of unsubstituted ZrCo2Sn is disrupted significantly by the Cu substitutions, leading to the breakdown of the magnetization versus electron count guidelines usually followed by Heusler phases, and a more typical metallic non-spin-polarized electronic structure at high x.