The classical Blech description of the electromigration - stress interplay in interconnects treats the maximum electromigration compressive stress as a material constant (the yield stress). Systematic drift velocity and hillock topography observations in bare Al, anodized Al, and Cu conductors show this description to be valid only for relatively low current densities j. With increasing j, the actual stress surpasses this threshold stress and increases progressively, i.e., the stress is shown to adjust to j. The explanation of this adjustment evolves naturally from the attribution of the stress relief mechanism to creep (time-dependent plastic flow). A diffusional creep mode is the one most likely to explain the data. Coble creep is ruled out as the diffusional creep mechanism, while Nabarro-Herring creep may be a good candidate. The thresholds for electromigration and for the egress of excess atoms diffusing to the hillock surface are addressed in terms of linear tensions of dislocations and surface ledges.