We analyse the time dependence of currents in a one-dimensional (1D) Bose gas in an optical lattice. For a 1D system, the stability of currents induced by accelerating the lattice exhibits a broad crossover as a function of the magnitude of the acceleration, and the strength of the inter-particle interactions. This differs markedly from mean-field results in higher dimensions. Using the infinite time evolving block decimation algorithm, we characterize this crossover by making quantitative predictions for the time-dependent behaviour of the currents and their decay rate. We also compute the time dependence of quasi-condensate fractions which can be measured directly in experiments. We compare our results to calculations based on phase-slip methods, finding agreement with the scaling as the particle density increases, but with significant deviations near unit filling.