Pruning neural networks, i.e., removing some of their parameters whilst retaining their accuracy, is one of the main ways to reduce the latency of a machine learning pipeline, especially in resource- and/or bandwidth-constrained scenarios. In this context, the pruning technique, i.e., how to choose the parameters to remove, is critical to the system performance. In this paper, we propose a novel pruning approach, called FlexRel and predicated upon combining training-time and inference-time information, namely, parameter magnitude and relevance, in order to improve the resulting accuracy whilst saving both computational resources and bandwidth. Our performance evaluation shows that FlexRel is able to achieve higher pruning factors, saving over 35% bandwidth for typical accuracy targets.