When the internal texture of a Gabor patch drifts orthogonally to its physical path, its perceived motion deviates dramatically from its physical path. The local position shifts accumulate to such an extent that a 45° oblique physical path appears to be vertical. However, at some point, a limit is reached and the path resets back to its veridical location, whereupon a new accumulation starts, making the new perceived path segment appear parallel to the pre-reset segment, but offset horizontally from it. Here, we tested whether spontaneous resets of this motion-induced position shift depend on the time or the distance over which position errors accrue, or both. We introduced a temporal gap in the middle of the path that forced the illusory path to reset back to its veridical physical position. This gap-triggered reset allowed us to measure the magnitude of the illusory offset up to that point. We found that perceived offset was less than expected for the angle of illusory drift, indicating that spontaneous resets had occurred prior to the gap-induced reset. The position offset decreased when the pre-gap duration increased but approximately doubled when the path length doubled. This pattern of perceived offsets is best accounted for by spontaneous resets that occur randomly over time at a constant rate, independently of the distance traveled. Our results suggest a temporal, not spatial, limit for the accumulation of position errors that underlies this illusion.