The number of devices in the Internet of Things (IoT) immensely grew in recent years. A frequent challenge in the assurance of the dependability of IoT systems is that components of the system appear as a black box. This paper presents a semi-automatic testing methodology for black-box systems that combines automata learning and fuzz testing. Our testing technique uses stateful fuzzing based on a model that is automatically inferred by automata learning. Applying this technique, we can simultaneously test multiple implementations for unexpected behavior and possible security vulnerabilities. We show the effectiveness of our learning-based fuzzing technique in a case study on the MQTT protocol. MQTT is a widely used publish/subscribe protocol in the IoT. Our case study reveals several inconsistencies between five different MQTT brokers. The found inconsistencies expose possible security vulnerabilities and violations of the MQTT specification.