Finite element analysis of computed tomography (CT) scans provides noninvasive estimates of bone strength at the spine and hip. To further validate such estimates clinically, we performed a 5‐year case‐control study of 1110 women and men over age 65 years from the AGES‐Reykjavik cohort (case = incident spine or hip fracture;​ control = no incident spine or hip fracture). From the baseline CT scans, we measured femoral and vertebral strength, as well as bone mineral density (BMD) at the hip (areal BMD only) and lumbar spine (trabecular volumetric BMD only). We found that for incident radiographically confirmed spine fractures (n = 167), the age‐adjusted odds ratio for vertebral strength was significant for women (2.8, 95% confidence interval [CI] 1.8 to 4.3) and men (2.2, 95% CI 1.5 to 3.2) and for men remained significant (p = 0.01) independent of vertebral trabecular volumetric BMD. For incident hip fractures (n = 171), the age‐adjusted odds ratio for femoral strength was significant for women (4.2, 95% CI 2.6 to 6.9) and men (3.5, 95% CI 2.3 to 5.3) and remained significant after adjusting for femoral neck areal BMD in women and for total hip areal BMD in both sexes;​ fracture classification improved for women by combining femoral strength with femoral neck areal BMD (p = 0.002). For both sexes, the probabilities of spine and hip fractures were similarly high at the BMD‐based interventional thresholds for osteoporosis and at corresponding preestablished thresholds for “fragile bone strength” (spine:​ women ≤ 4500 N, men ≤ 6500 N;​ hip:​ women ≤ 3000 N, men ≤ 3500 N). Because it is well established that individuals over age 65 years who have osteoporosis at the hip or spine by BMD criteria should be considered at high risk of fracture, these results indicate that individuals who have fragile bone strength at the hip or spine should also be considered at high risk of fracture.