We report the magnetic structure of two of the magnetically ordered phases of Co-3(OH)(2)(C4O4)2 center dot 3H(2)O, a coordination polymer that consists of a triangular framework decorated with anisotropic Co(II) ions. The combination of neutron diffraction experiments and magnetic susceptibility data allows us to identify one phase as displaying spin idle behavior, where only a fraction of the moments order at intermediate temperatures, while at the lowest temperatures the system orders fully. This novel magnetic behavior is discussed within the framework of a simple Hamiltonian and representational analysis and rationalizes this multiphase behavior by considering the combination of frustration and anisotropy.