Decoherence phenomenona in a neutron interferometer are analyzed by simulation of the effects of an environment with magnetic noise fields. Basic calculations and experiments show the validity and limitations of this model system. In particular, loss and recovery of the interference pattern with controllable noise sources in both interferometer arms are discussed in detail. In addition, the decoherence behavior at high interference order, where Schrödinger-cat-like states exist in the interferometer, is investigated. While at low interference order a smearing of the interference pattern is observed, at high interference order a smearing of the modulated momentum distribution occurs.