The vehicle routing problem with time windows (VRPTW) is a classical problem with many applications. To improve schedules accuracy, the models may consider time dependency in travel time. The real traffic data used as input to the optimizers can be obtained by location providers that predict the travel times for specified timeframes. The predictors estimate travel time based on historical data, where uncertainties from accidents, weather conditions, and working roads, may add unpredictable delays. These providers enhance their forecasts by monitoring the live traffic and updating the travel time throughout the day. Since routes are typically designed in advance, updates are hard to integrate once execution starts. In this context, this work introduces the robust time-dependent VRPTW (RTDVRPTW), which uses robust optimization to handle uncertainty in travel times. In this paper, we discuss the computational properties of RTDVRPTW under various assumptions, present innovative exact methods, and report results from real-data experiments using an efficient heuristic approach to offer practical insights.