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We introduce a new machine-learning-based approach, which we call the Independent Classifier networks (InClass nets) technique, for the nonparameteric estimation of conditional independence mixture models (CIMMs). We approach the estimation of a CIMM as a multi-class classification problem, since dividing the dataset into different categories naturally leads to the estimation of the mixture model. InClass nets consist of multiple independent classifier neural networks (NNs), each of which handles one of the variates of the CIMM. Fitting the CIMM to the data is performed by simultaneously training the individual NNs using suitable cost functions. The ability of NNs to approximate arbitrary functions makes our technique nonparametric. Further leveraging the power of NNs, we allow the conditionally independent variates of the model to be individually high-dimensional, which is the main advantage of our technique over existing non-machine-learning-based approaches. We derive some new results on the nonparametric identifiability of bivariate CIMMs, in the form of a necessary and a (different) sufficient condition for a bivariate CIMM to be identifiable. We provide a public implementation of InClass nets as a Python package called RainDancesVI and validate our InClass nets technique with several worked out examples. Our method also has applications in unsupervised and semi-supervised classification problems.