We present dynamical models of four interacting systems: NGC 5257/8, The Mice, the Antennae, and NGC 2623. The parameter space of the encounters are constrained using the Identikit model-matching and visualization tool. Identikit utilizes hybrid N-body and test particle simulations to enable rapid exploration of the parameter space of galaxy mergers. The Identikit-derived matches of these systems are reproduced with self-consistent collisionless simulations which show very similar results. The models generally reproduce the observed morphology and H I kinematics of the tidal tails in these systems with reasonable properties inferred for the progenitor galaxies. The models presented here are the first to appear in the literature for NGC 5257/8 and NGC 2623, and The Mice and the Antennae are compared with previously published models. Based on the assumed mass model and our derived initial conditions, the models indicate that the four systems are currently being viewed 175-260 Myr after first passage and cover a wide range of merger stages. In some instances there are mismatches between the models and the data (e.g., in the length of a tail); these are likely due to our adoption of a single mass model for all galaxies. Despite the use of a single mass model, these results demonstrate the utility of Identikit in constraining the parameter space for galaxy mergers when applied to real data.