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A key but unresolved issue in the study of human mortality at older ages is whether mortality is being compressed (which implies that we may be approaching a maximum limit to the length of life) or postponed (which would imply that we are not). We analyze historical and current population mortality data between ages 50 and 100 by birth cohort in 19 currently-industrialized countries, using a Bayesian technique to surmount cohort censoring caused by survival, to show that while the dominant historical pattern has been one of mortality compression, there have been occasional episodes of mortality postponement. The pattern of postponement and compression across different birth cohorts explain why longevity records have been slow to increase in recent years: we find that cohorts born between around 1900 and 1950 are experiencing historically unprecedented mortality postponement, but are still too young to break longevity records. As these cohorts attain advanced ages in coming decades, longevity records may therefore increase significantly. Our results confirm prior work suggesting that if there is a maximum limit to the human lifespan, we are not yet approaching it.