Appointment scheduling problems under uncertainty encounter a fundamental trade-off between cost minimization and customer waiting times. Most existing studies address this trade-off using a weighted sum approach, which puts little emphasis on individual waiting times and, thus, customer satisfaction. In contrast, we study how to minimize total cost while providing waiting time guarantees to all customers. Given box uncertainty sets for service times and no-shows, we introduce the Robust Appointment Scheduling Problem with Waiting Time Guarantees. We show that the problem is NP-hard in general and introduce a mixed-integer linear program that can be solved in reasonable computation time. For special cases, we prove that polynomial-time variants of the well-known Smallest-Variance-First sequencing rule and the Bailey-Welch scheduling rule are optimal. Furthermore, a case study with data from the radiology department of a large university hospital demonstrates that the approach not only guarantees acceptable waiting times but, compared to existing robust approaches, may simultaneously reduce costs incurred by idle time and overtime. This work suggests that limiting instead of minimizing customer waiting times is a win-win solution in the trade-off between customer satisfaction and cost minimization. Additionally, it provides an easy-to-implement and customizable appointment scheduling framework with waiting time guarantees.