Brueckner coupled cluster (B-CC) methods have seen a considerable rise in popularity over the last decade thanks, in part, to their apparent propensity for avoiding artifactual symmetry-breaking problems that sometimes plague Hartree-Fock-based approaches. Recent B-CC applications to problematic systems such as the tetraoxygen cation have provided encouraging examples of the success of this theory. In the present work, we examine the performance of the Brueckner technique for a number of other well-known symmetry-breaking problems, including the formyloxyl radical, the first excited state of NO2 and the nitrate radical. In these cases, B-CC methods are found to fail dramatically, predicting broken-symmetry equilibrium geometries in conflict with experimental and/or higher-level theoretical results. A framework is developed which indicates that these errors can be attributed to artificially exaggerated second-order Jahn-Teller interactions with nearby electronic states. Hence, in spite of their initial successes, Brueckner methods cannot be considered a panacea for symmetry-breaking problems.