  We investigate the consistency of the background-field formalism when
applying various regularizations and renormalization schemes. By an example of
a two-dimensional $\sigma$ model it is demonstrated that the background-field
method gives incorrect results when the regularization (and/or renormalization)
is noninvariant. In particular, it is found that the cut-off regularization and
the differential renormalization belong to this class and are incompatible with
the background-field method in theories with nonlinear symmetries.
