High-cardinality categorical features are a major challenge for machine learning methods in general and for deep learning in particular. Existing solutions such as one-hot encoding and entity embeddings can be hard to scale when the cardinality is very high, require much space, are hard to interpret or may overfit the data. A special scenario of interest is that of repeated measures, where the categorical feature is the identity of the individual or object, and each object is measured several times, possibly under different conditions (values of the other features). We propose accounting for high-cardinality categorical features as random effects variables in a regression setting, and consequently adopt the corresponding negative log likelihood loss from the linear mixed models (LMM) statistical literature and integrate it in a deep learning framework. We test our model which we call LMMNN on simulated as well as real datasets with a single categorical feature with high cardinality, using various baseline neural networks architectures such as convolutional networks and LSTM, and various applications in e-commerce, healthcare and computer vision. Our results show that treating high-cardinality categorical features as random effects leads to a significant improvement in prediction performance compared to state of the art alternatives. Potential extensions such as accounting for multiple categorical features and classification settings are discussed. Our code and simulations are available at https://github.com/gsimchoni/lmmnn.