The magnetization changes of a ferromagnet in a slowly varying applied field typically occur in a sequence of abrupt jumps, or avalanches, displaying a wide range of size and time scales (Barkhausen noise). This behavior has been modeled as a rigid magnetic domain wall moving in a disordered medium, and as a self-organizing process involving many interacting degrees of freedom (flexible domain wall). Our investigation of the avalanche sequence in an Fe-Ni-Co alloy has revealed features that are almost exactly reproducible (events that occur at the same field on every cycle) intermingled with events that show no apparent reproduciblity. The behavior is temperature and driving rate independent, indicating that dynamical effects involving multiple degrees of freedom are taking place. An analysis of correlations within the avalanche sequence also suggests that the rigid domain wall model is inadequate. The correlations also reveal the effect of long-range magnetic interactions. A simple model of domain wall motion, when modified to include long-range interactions, exhibits self-organized criticality.