Network dynamical systems are ubiquitous in science and engineering. The most basic property of a network dynamical system is its size, which, for scalar dynamics, corresponds to the number of nodes. For linear network systems, recent studies have developed reliable tools for inferring the size of the system from perceptible dynamics (measurements of one or some of the network nodes) across multiple experiments. Here, we extend these tools to nonlinear network systems by putting forward a model-agnostic approach that combines clustering techniques, detection matrices, and spectral analysis. The theoretical premise of the algorithm is that, under mild assumptions, the variation between the dynamics of some nodes across multiple measurements can be used to bound the variation between the dynamics of all nodes across the same measurements.