The line tension Gamma of a dislocation is an important and fundamental property ubiquitous to continuum scale models of metal plasticity. However, the precise value of Gamma in a given material has proven difficult to assess, with literature values encompassing a wide range. Here results from a multiscale simulation and robust analysis of the dislocation line tension, for dislocation bow-out between pinning points, are presented for two widely-used interatomic potentials for Al. A central part of the analysis involves an effective Peierls stress applicable to curved dislocation structures that markedly differs from that of perfectly straight dislocations but is required to describe the bow-out both in loading and unloading. The line tensions for the two interatomic potentials are similar and provide robust numerical values for Al. Most importantly, the atomic results show notable differences with singular anisotropic elastic dislocation theory in that (i) the coefficient of the ln(L) scaling with dislocation length L differs and (ii) the ratio of screw to edge line tension is smaller than predicted by anisotropic elasticity. These differences are attributed to local dislocation core interactions that remain beyond the scope of elasticity theory. The many differing literature values for Gamma are attributed to various approximations and inaccuracies in previous approaches. The results here indicate that continuum line dislocation models, based on elasticity theory and various core-cut-off assumptions, may be fundamentally unable to reproduce full atomistic results, thus hampering the detailed predictive ability of such continuum models.