We consider the optimization problem associated with training simple ReLU neural networks of the form $\mathbf{x}\mapsto \sum_{i=1}^{k}\max\{0,\mathbf{w}_i^\top \mathbf{x}\}$ with respect to the squared loss. We provide a computer-assisted proof that even if the input distribution is standard Gaussian, even if the dimension is arbitrarily large, and even if the target values are generated by such a network, with orthonormal parameter vectors, the problem can still have spurious local minima once $6\le k\le 20$. By a concentration of measure argument, this implies that in high input dimensions, \emph{nearly all} target networks of the relevant sizes lead to spurious local minima. Moreover, we conduct experiments which show that the probability of hitting such local minima is quite high, and increasing with the network size. On the positive side, mild over-parameterization appears to drastically reduce such local minima, indicating that an over-parameterization assumption is necessary to get a positive result in this setting.