Deep neural networks (DNNs) demonstrate outstanding performance across most computer vision tasks. Some critical applications, such as autonomous driving or medical imaging, also require investigation into their behavior and the reasons behind the decisions they make. In this vein, DNN attribution consists in studying the relationship between the predictions of a DNN and its inputs. Attribution methods have been adapted to highlight the most relevant weights or neurons in a DNN, allowing to more efficiently select which weights or neurons can be pruned. However, a limitation of these approaches is that weights are typically compared within each layer separately, while some layers might appear as more critical than others. In this work, we propose to investigate DNN layer importance, i.e. to estimate the sensitivity of the accuracy w.r.t. perturbations applied at the layer level. To do so, we propose a novel dataset to evaluate our method as well as future works. We benchmark a number of criteria and draw conclusions regarding how to assess DNN layer importance and, consequently, how to budgetize layers for increased DNN efficiency (with applications for DNN pruning and quantization), as well as robustness to hardware failure (e.g. bit swaps).