We examine an exactly solvable model of decoherence --- a spin-system interacting with a collection of environment spins. We show that in this simple model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths typically lead to a non-Markovian (Gaussian) suppression of coherence between pointer states. We explore the regime of validity of this result and discuss its relation to spectral features of the environment. We also consider its relevance to Loschmidt echo experiments (which measure, in effect, the fidelity between the initial state and the state first evolved forward with a Hamiltonian {H}, and then ``unevolved'' with (approximately) {-H}). In particular, we show that for partial reversals (e.g. , when only a part of the total Hamiltonian changes sign) fidelity may exhibit a Gaussian dependence on the time of reversal that is independent of the details of the reversal procedure: It just depends on what part of the Hamiltonian gets ``flipped'' by the reversal. This puzzling behavior was observed in several NMR experiments. Natural candidates for such two environments (one of which is easily reversed, while the other is ``irreversible'') are suggested for the experiment involving ferrocene.

PACS numbers: 03.65.Yz, 03.67.--a