The magnetic properties of nonsuperconducting (${\mathrm{Pr}}_{1.5}$ ${\mathrm{Ce}}_{0.5}$ )${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{2}$ ${\mathrm{MO}}_{10\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ with M=Nb, Ta are characterized with dc magnetization, specific-heat, and neutron-diffraction experiments. Data for (${\mathrm{Pr}}_{1.5}$ ${\mathrm{Ce}}_{0.5}$ )${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{2}$ ${\mathrm{NbO}}_{10\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ reveal complex Cu magnetism marked by antiferromagnetic order below 200 K, spin structure transitions at 130 and 57 K, both collinear and noncollinear antiferromagnetic spin structures, and weak ferromagnetic behavior below 130 K. The data also indicate an anomalous ordering of the Pr spins near 10 K, a large linear contribution to the low-temperature specific heat, and a Pr 4f crystal-field ground state similar to that found in ${\mathrm{PrBa}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{7}$ . Furthermore, there is evidence that the weak ferromagnetic behavior couples to the Pr ordering near 10 K. Identical Pr magnetism and similar Cu magnetism are found in (${\mathrm{Pr}}_{1.5}$ ${\mathrm{Ce}}_{0.5}$ )${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{2}$ ${\mathrm{TaO}}_{10\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ , deoxygenated (${\mathrm{Pr}}_{1.5}$ ${\mathrm{Ce}}_{0.5}$ )${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{2}$ ${\mathrm{NbO}}_{10\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ , and deoxygenated (${\mathrm{Pr}}_{1.5}$ ${\mathrm{Ce}}_{0.5}$ )${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{2}$ ${\mathrm{TaO}}_{10\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ . These results indicate that superconductivity is suppressed in these compounds in the same phenomenological manner as in ${\mathrm{PrBa}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{7}$ . We interpret this as evidence that superconductivity is suppressed by the same mechanism in both structures and propose that a general correlation exists between anomalous Pr magnetism and a lack of superconductivity in these Pr-based high-${\mathrm{T}}_{\mathrm{C}}$ cuprates. The significance of these results and analyses to understanding and modeling the suppression of superconductivity by Pr in high-${\mathrm{T}}_{\mathrm{C}}$ cuprates is discussed.