In this paper, we leverage two large-scale real-world datasets to provide the first results on the limits of predictability of cellular data traffic demands generated by individual users over time and space. Using information theory tools, we measure the maximum predictability that any algorithm has potential to achieve. We first focus on the predictability of mobile traffic consumption patterns in isolation. Our results show that it is theoretically possible to anticipate the individual demand with a typical accuracy of 85% and reveal that this percentage is consistent across all user types. Then, we analyze the joint predictability of the traffic demands and mobility patterns. We find that the two dimensions are correlated, which improves the predictability upper bound to 90% on average.