Is the Anthropocene a new geological epoch and can its beginning be determined? The so-called Great Acceleration data, used repeatedly in support of the concept of the Anthropocene as a new geological epoch, are closely examined. They are supposed to be characterised by a clear intensification of growth (sharp increase, acceleration) in the mid-20th century. They revealed the opposite effect: they are characterised by decelerations either at precisely the same time when accelerations were expected or over the entire range of data, the phenomenon described here as the Great Deceleration. Implications of this study are two-folds: (1) contrary to expectations, these data cannot be used to determine the beginning of the Anthropocene and (2) the common deceleration questions whether humans alone can cause a transition to a new geological epoch. Investigation of the growth of the genus Homo in the past 2,000,000 years suggests that what is now called the Anthropocene could be just a natural continuation of the gradual evolution of human activities and impacts over a long time, without a sudden intensification and without a convincing evidence for a transition to a new geological epoch.