T1-weighted structural MRI is widely used to measure brain morphometry (e.g., cortical thickness and subcortical volumes). Accelerated scans as fast as one minute or less are now available but it is unclear if they are adequate for quantitative morphometry. Here we compared the measurement properties of a widely adopted 1.0 mm resolution scan from the Alzheimer's Disease Neuroimaging Initiative (ADNI = 5'12'') with two variants of highly accelerated 1.0 mm scans (compressed-sensing, CSx6 = 1'12''; and wave-controlled aliasing in parallel imaging, WAVEx9 = 1'09'') in a test-retest study of 37 older adults aged 54 to 86 (including 19 individuals diagnosed with a neurodegenerative dementia). Rapid scans produced highly reliable morphometric measures that largely matched the quality of morphometrics derived from the ADNI scan. Regions of lower reliability and relative divergence between ADNI and rapid scan alternatives tended to occur in midline regions and regions with susceptibility-induced artifacts. Critically, the rapid scans yielded morphometric measures similar to the ADNI scan in regions of high atrophy. The results converge to suggest that, for many current uses, extremely rapid scans can replace longer scans. As a final test, we explored the possibility of a 0'49'' 1.2 mm CSx6 structural scan, which also showed promise. Rapid structural scans may benefit MRI studies by shortening the scan session and reducing cost, minimizing opportunity for movement, creating room for additional scan sequences, and allowing for the repetition of structural scans to increase precision of the estimates.