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The breakdown point plays an important though at times controversial role in statistics. In situations in which it has proved most successful there is a group of transformations which act on the sample space and which give rise to an equivariance structure. For equivariant functionals, that is those functionals which respect the group structure, a non-trivial upper bound for the breakdown point was derived in Davies and Gather (2005). The present paper briefly repeats the main results of Davies and Gather (2005) but is mainly concerned with giving additional insight into the concept of breakdown point. In particular, we discuss the attainability of the bound and the dependence of the breakdown point on the sample or distribution and on the metrics used in its definition.