  We explore the connection between two recently introduced notions of
non-Markovian quantum dynamics and the validity of the so-called quantum
regression theorem. While non-Markovianity of a quantum dynamics has been
defined looking at the behaviour in time of the statistical operator, which
determines the evolution of mean values, the quantum regression theorem makes
statements about the behaviour of system correlation functions of order two and
higher. The comparison relies on an estimate of the validity of the quantum
regression hypothesis, which can be obtained exactly evaluating two points
correlation functions. To this aim we consider a qubit undergoing dephasing due
to interaction with a bosonic bath, comparing the exact evaluation of the
non-Markovianity measures with the violation of the quantum regression theorem
for a class of spectral densities. We further study a photonic dephasing model,
recently exploited for the experimental measurement of non-Markovianity. It
appears that while a non-Markovian dynamics according to either definition
brings with itself violation of the regression hypothesis, even Markovian
dynamics can lead to a failure of the regression relation.
