The application of function fitting neural networks in microstructural MRI has so far been restricted to lower-dimensional biophysical models. Moreover, the data sufficiency requirements of learning-based approaches remain unclear. Here, we use supervised learning to vastly accelerate the fitting of a high-dimensional relaxation-diffusion model of tissue microstructure and develop analysis tools for assessing the accuracy and sensitivity of model fitting networks. The developed learning-based fitting pipelines were tested on relaxation-diffusion data acquired with optimal and sub-optimal protocols. We found no evidence that machine-learning algorithms can correct for a degenerate fitting landscape or replace a careful design of the acquisition protocol.