Prior experiences can influence future actions. These experiences can not only drive adaptive changes in motor output, but they can also modulate the rate at which these adaptive changes occur. Here we studied anterograde interference in motor adaptation--the ability of a previously learned motor task (Task A) to reduce the rate of subsequently learning a different (and usually opposite) motor task (Task B). We examined the formation of the motor systems capacity for anterograde interference in the adaptive control of human reaching-arm movements by determining the amount of interference after varying durations of exposure to Task A (13, 41, 112, 230, and 369 trials). We found that the amount of anterograde interference observed in the learning of Task B increased with the duration of Task A. However, this increase did not continue indefinitely, instead, the interference reached asymptote after 15-40 trials of Task A. Interestingly, we found that a recently proposed multi-rate model of motor adaptation, composed of two distinct but interacting adaptive processes, predicts several key features of the interference patterns we observed. Specifically, this computational model (without any free parameters) predicts the initial growth and leveling off of anterograde interference that we describe, as well as the asymptotic amount of interference that we observe experimentally (R(2) = 0.91). Understanding the mechanisms underlying anterograde interference in motor adaptation may enable the development of improved training and rehabilitation paradigms that mitigate unwanted interference.