We present a model to describe the concentration-dependent growth of protein filaments. Our model contains two states, a low-entropy/high-affinity ordered state and a high-entropy/low-affinity disordered state. Consistent with experiments, our model shows a diffusion-limited linear growth regime at low concentration, followed by a concentration-independent plateau at intermediate concentrations, and rapid disordered precipitation at the highest concentrations. We show that growth in the linear and plateau regions is the result of two processes that compete amid the rapid binding and unbinding of nonspecific states. The first process is the addition of ordered molecules during periods in which the end of the filament is free of incorrectly bound molecules. The second process is the capture of defects, which occurs when consecutive ordered additions occur on top of incorrectly bound molecules. We show that a key molecular property is the probability that a diffusive collision results in a correctly bound state. Small values of this probability suppress the defect capture growth mode, resulting in a plateau in the growth rate when incorrectly bound molecules become common enough to poison ordered growth. We show that conditions that nonspecifically suppress or enhance intermolecular interactions, such as the addition of depletants or osmolytes, have opposite effects on the growth rate in the linear and plateau regimes. In the linear regime, stronger interactions promote growth by reducing dissolution events, but in the plateau regime stronger interactions inhibit growth by stabilizing incorrectly bound molecules.