What kinds of systems have so far failed to be modeled via supervised artificial network training? Artificial networks model systems with a set of inputs and outputs and expected behavior. To train a network for modeling such systems, hundreds, thousands, or millions of example inputs-output pairs may be required. This is called a labelled data set, and the network and its optimization algorithm are meant to find a set of network parameters that best match the I/O of the artificial network with the I/O of the system.

Are there any systems, for which sufficient labelled data sets exist, that have yet to be successfully modeled with artificial networks of any type (recurrent, deep, convolution, etc)?
I haven't seen any dataset where some standard models worked and neural networks utterly failed.

For columnar data (e.g. Excel files / database dumps / CSV files) which contain structured data usually tree-based models like random forests and gradient boosting work better, but neural networks are also usually way better than random.

If you demand other things, e.g. explanations for the decision then Bayesian models might give you an easier time. Or for baselines/simple implementations linear models. Or for real time applications...