We leverage two large-scale real-world datasets to provide the pioneer results on the limits of predictability of per-user mobile data traffic demands 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 data traffic consumption patterns in isolation. Our results show that it is theoretically possible to anticipate individual demands with a typical accuracy of 85%. Then, we analyze the joint predictability of mobile data traffic demands and mobility patterns. Their correlation that we find leads to a higher theoretical potential performance in joint prediction. Besides, we propose a novel practice to evaluate the spatiotemporal correlation of per-user mobile data traffic.