  We study the interplay between forgetful and memory-keeping evolution
enforced on a two-level system by a multi-spin environment whose elements are
coupled to local bosonic baths. Contrarily to the expectation that any
non-Markovian effect would be buried by the forgetful mechanism induced by the
spin-bath coupling, one can actually induce a full Markovian-to-non-Markovian
transition of the two-level system's dynamics, controllable by parameters such
as the mismatch between the energy of the two-level system and of the spin
environment. For a symmetric coupling, the amount of non-Markovianity
surprisingly grows with the number of decoherence channels.
