Soil salinity and sodicity are major constraints to global cereal production, but breeding for tolerance has been slow. Narrow gene pools, over-emphasis on the sodium (Na+) exclusion mechanism, little attention to osmotic stress/tissue tolerance mechanism(s) in which accumulation of inorganic ions such as Na+ is implicated, and lack of a suitable screening method have impaired progress. The aims of this study were to discover novel genes for Na+ accumulation using genome-wide association studies, compare growth responses to salinity and sodicity in low-Na+ bread Westonia with Nax1 and Nax2 genes and high-Na+ bread wheat Baart-46, and evaluate growth responses to salinity and sodicity in bread wheats with varying leaf Na+ concentrations. The novel high-Na+ bread wheat germplasm, MW#293, had higher grain yield under salinity and sodicity, in absolute and relative terms, than the other bread wheat entries tested. Genes associated with high Na+ accumulation in bread wheat were identified, which may be involved in tissue tolerance/osmotic adjustment. As most modern bread wheats are efficient at excluding Na+, further reduction in plant Na+ is unlikely to provide agronomic benefit. The salinity and sodicity tolerant germplasm MW#293 provides an opportunity for the development of future salinity/sodicity tolerant bread wheat.