  A cancellation of conformal anomalies in d=4 for the $C^2+\bar C^2$ and of
chiral anomalies for the $C^2-\bar C^2$ is known in N-extended supergravities
with $N \geq 5$. We propose an explanation of these cancellations using
supersymmetry and dimension of linearized chiral superfields. We contrast these
models with the ones with $N < 5$, where both types of anomalies are known to
be present.
