We show, using a large new data set, that the temporally resolved speed of active cargo transport in living cells follows a scaling law over several decades of time and length. The statistical regularities display a time-averaged shape that we interpret to reflect stress buildup, followed by rapid release. The scaling power law agrees quantitatively with those reported in inanimate systems (jammed colloids and granular media, and magnetic Barkhausen noise), suggesting a common origin in pushing through a crowded environment in a weak force regime. The implied regulation of the speed of active cellular transport due to environmental obstruction results in bursts of speed and acceleration. These findings extend the classical notion of molecular crowding.