Until recently human longevity records continued to grow in history, with no indication of approaching a hypothetical longevity limit. Also, earlier studies found that age-specific death rates cease to increase at advanced ages (mortality plateau) suggesting the absence of fixed limit to longevity too. In this study, we reexamine both claims with more recent and reliable data on supercentenarians (persons aged 110 years and older). We found that despite a dramatic historical increase in the number of supercentenarians, further growth of human longevity records in subsequent birth cohorts slowed down significantly and almost stopped for those born after 1879. We also found an exponential acceleration of age-specific death rates for persons older than 113 years in more recent data. Slowing down the historical progress in maximum reported age at death and accelerated growth of age-specific death rates after age 113 years in recent birth cohorts may indicate the need for more conservative estimates for future longevity records unless a scientific breakthrough in delaying aging would happen. The hypothesis of approaching a biological limit to human longevity has received some empirical support and it deserves further study and testing.