  The empirical spectra and $E2$ decay rates in $^{110,112,114,116}$Cd are
shown to be consistent with a vibrational interpretation for low-lying normal
states, coexisting with a single deformed $\gamma$-soft band of intruder
states. The observed deviations from this paradigm show up in particular
non-yrast states, which are properly described by a Hamiltonian with U(5)
partial dynamical symmetry. The latter is characterized by a good (broken)
symmetry in most (in selected) normal states, weakly coupled to intruder
states.
