We present a stochastic model of filament growth driven by the motor-assisted transport of particles along the filament. We show how the growth can be analyzed in terms of a sequence of first passage times for a particle hopping between the two ends of the filament, and use this to calculate the mean and variance of the length as a function of time. We determine how the growth depends on the waiting time density of the underlying hopping process, and highlight differences in the growth generated by normal and anomalous transport, for which the mean waiting time is finite and infinite, respectively. In the case of normal transport, we determine the length at which there is a balance between particle-driven assembly and particle-independent disassembly of the filament. The existence of such a balance point is thought to provide a mechanism for flagellar length control.