Grid cells in the rodent entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space, and have been hypothesized to form the neural substrate for dead-reckoning. However, in previous models suggested for grid cell activity, errors accumulate rapidly in the integration of velocity inputs. To produce grid-cell like responses, these models would require frequent resets triggered by external sensory cues, casting doubt on the dead-reckoning potential of the grid cell system. Here we focus on the accuracy of path integration in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat's velocity and heading. We consider the role of the network boundary in integration performance, and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We show that the rate at which errors accumulate in the integration of velocity depends on the network's organization and size, and on the intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model can accurately integrate velocity inputs over a maximum of ~10-100 m and ~1-10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in dMEC. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.