This paper deals with learnability of concept classes defined by neural networks, showing the hardness of PAC-learning (in the complexity, not merely information-theoretic sense) for networks with a particular class of activation. The obstruction lies not with the VC dimension, which is known to grow slowly; instead, the result follows the fact that the loading problem is NP-complete. (The complexity scales badly with input dimension; the loading problem is polynomial-time if the input dimension is constant). Similar and well-known theorems had already been proved by Megiddo and by Blum and Rivest, for binary-threshold networks. It turns out the general problem for continuous sigmoidal-type functions, as used in practical applications involving steepest descent, is not NP-hard – there are “sigmoidals” for which the problem is in fact trivial – so it is an open question to determine what properties of the activation function cause difficulties. Ours is the first result on the hardness of loading networks which do not consist of binary neurons; we employ a piecewise-linear activation function that has been used in the neural network literature. Our theoretical results lend further justification to the use of incremental (architecture-changing) techniques for training networks.