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We present a modular Bayesian optimization framework that efficiently generates time-optimal trajectories for a cooperative multi-agent system, such as a team of UAVs. Existing methods for multi-agent trajectory generation often rely on overly conservative constraints to reduce the complexity of this high-dimensional planning problem, leading to suboptimal solutions. We propose a novel modular structure for the Bayesian optimization model that consists of multiple Gaussian process surrogate models that represent the dynamic feasibility and collision avoidance constraints. This modular structure alleviates the stark increase in computational cost with problem dimensionality and enables the use of minimal constraints in the joint optimization of the multi-agent trajectories. The efficiency of the algorithm is further improved by introducing a scheme for simultaneous evaluation of the Bayesian optimization acquisition function and random sampling. The modular BayesOpt algorithm was applied to optimize multi-agent trajectories through six unique environments using multi-fidelity evaluations from various data sources. It was found that the resulting trajectories are faster than those obtained from two baseline methods. The optimized trajectories were validated in real-world experiments using four quadcopters that fly within centimeters of each other at speeds up to 7.4 m/s.