The combination of plate tectonic models of intraplate stress and observations of the directions of principal midplate stresses provides two types of constraints on the magnitude of deviatoric stress in the lithosphere. First, the fit of global tectonic stress models to observations can constrain stress magnitudes if the magnitude of one of the component driving forces can be independently estimated. Second, if in the vicinity of a calculable local source of stress the observations of stress orientations indicate comparable control by the local and regional stress field, then the magnitude of the regional field may be estimated. Global intraplate stress models have been calculated using a finite element technique for a wide variety of possible driving force combinations. The best fitting global stress models include ridge pushing forces as an essential element and have deviatoric stress magnitudes comparable to the horizontal compressive stress exerted by ridge elevation, estimated on independent grounds to be 200--300 bar. Regional stresses in the Indian plate are shown on the basis of midplate earthquake mechanisms to be larger than the ~100-bar additional compressive stress exerted by the isostatically compensated topography of the Ninetyeast Ridge. In the vicinity of Hawaii, however, plate bending stresses of perhaps 1-kbar or greater magnitude may dominate regional stress. An average level of several hundred bars for the magnitude of typical intraplate deviatoric stresses is consistent with these various constraints.