Network slicing is a new paradigm for future 5G networks where the network infrastructure is divided into slices devoted to different services and customized to their needs. With this paradigm, it is essential to allocate to each slice the needed resources, which requires the ability to forecast their respective demands. To this end, we present DeepCog, a novel data analytics tool for the cognitive management of resources in 5G systems. DeepCog forecasts the capacity needed to accommodate future traffic demands within individual network slices while accounting for the operator's desired balance between resource overprovisioning (i.e., allocating resources exceeding the demand) and service request violations (i.e., allocating less resources than required). To achieve its objective, DeepCog hinges on a deep learning architecture that is explicitly designed for capacity forecasting. Comparative evaluations with real-world measurement data prove that DeepCog's tight integration of machine learning into resource orchestration allows for substantial (50% or above) reduction of operating expenses with respect to resource allocation solutions based on state-of-the-art mobile traffic predictors. Moreover, we leverage DeepCog to carry out an extensive first analysis of the trade-off between capacity overdimensioning and unserviced demands in adaptive, sliced networks and in presence of real-world traffic.