Weak gravitational lensing has become an important method to determine the masses of galaxy clusters. The intrinsic shapes of the galaxies are a dominant source of uncertainty, but there are other limitations to the precision that can be achieved. In this paper we revisit a typically ignored source of uncertainty: uncorrelated structure along the line of sight. Using results from the Millennium Simulation we confirm the validity of analytical calculations that have shown that such random projections are particularly important for studies of the cluster density profile. In general the contribution of distant large-scale structure to the total error budget is comparable to the statistical errors. We find that the precision of the mass measurement can be improved only slightly by modelling the uncorrelated large-scale structure using readily available data.