Objective: We show that state-of-the-art deep neural networks achieve superior results in regression-based multi-class proportional myoelectric hand prosthesis control than two common baseline approaches, and we analyze the neural network mapping to explain why this is the case. Methods: Feedforward neural networks and baseline systems are trained on an offline corpus of 11 able-bodied subjects and 4 prosthesis wearers, using the R2 score as metric. Analysis is performed using diverse qualitative and quantitative approaches, followed by a rigorous evaluation. Results: Our best neural networks have at least three hidden layers with at least 128 neurons per layer; smaller architectures, as used by many prior studies, perform substantially worse. The key to good performance is to both optimally regress the target movement, and to suppress spurious movements. Due to the properties of the underlying data, this is impossible to achieve with linear methods, but can be attained with high exactness using sufficiently large neural networks. Conclusion: Neural networks perform significantly better than common linear approaches in the given task, in particular when sufficiently large architectures are used. This can be explained by salient properties of the underlying data, and by theoretical and experimental analysis of the neural network mapping. Significance: To the best of our knowledge, this work is the first one in the field which not only reports that large and deep neural networks are superior to existing architectures, but also explains this result.