In an increasing number of applications designers have access to multiple computer models which typically have different levels of fidelity and cost. Traditionally, designers calibrate these models one at a time against some high-fidelity data (e.g., experiments). In this paper, we question this tradition and assess the potential of calibrating multiple computer models at the same time. To this end, we develop a probabilistic framework that is founded on customized neural networks (NNs) that are designed to calibrate an arbitrary number of computer models. In our approach, we (1) consider the fact that most computer models are multi-response and that the number and nature of calibration parameters may change across the models, and (2) learn a unique probability distribution for each calibration parameter of each computer model, (3) develop a loss function that enables our NN to emulate all data sources while calibrating the computer models, and (4) aim to learn a visualizable latent space where model-form errors can be identified. We test the performance of our approach on analytic and engineering problems to understand the potential advantages and pitfalls in simultaneous calibration of multiple computer models. Our method can improve predictive accuracy, however, it is prone to non-identifiability issues in higher-dimensional input spaces that are normally constrained by underlying physics.