  We investigate some peculiarities in the calculation of the two-loop
beta-function of $N=1$ supersymmetric models which are intimately related to
the so-called "Anomaly Puzzle". There is an apparent paradox when the
computation is performed in the framework of the covariant derivative
background field method. In this formalism, it is obtained a finite two-loop
effective action, although a non-null coefficient for the beta-function is
achieved by means of the renormalized two-point function in the background
field. We show that if the standard background field method is used, this
two-point function has a divergent part which allows for the calculation of the
beta-function via the renormalization constants, as usual. Therefore, we
conjecture that this paradox has its origin in the covariant supergraph
formalism itself, possibly being an artifact of the rescaling anomaly.
