We investigate the reliability of transition strengths computed in the random-phase approximation (RPA), comparing with exact results from diagonalization in full 0ħω shell-model spaces. The RPA and shell-model results are in reasonable agreement for most transitions; however, some very low lying collective transitions, such as isoscalar quadrupole ones, are in serious disagreement. We suggest that the failure lies with incomplete restoration of broken symmetries in the RPA. Furthermore, we prove, analytically and numerically, that standard statements regarding the energy-weighted sum rule in the RPA do not hold if an exact symmetry is broken.