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A systematic study of the persistent current decay and snapback effect in the fields of Tevatron accelerator dipoles was performed at the Fermilab Magnet Test Facility (MTF). The decay and snapback were measured under a range of conditions including variations of the current ramp parameters and magnet operational history. The study has mostly focused on the dynamic behavior of the normal sextupole component. In addition, the paper presents the persistent current effects observed in the other allowed field harmonics as well. The results provide new information about the previously observed "excess" decay during the first several seconds of the sextupole decay during injection and the correlation between the snapback amplitude and its duration.