Recently, unmanned aerial vehicle (UAV)-oriented applications have been growing worldwide. Thus, there is a strong interest in using UAVs for applications requiring wide-area connectivity coverage. Such applications might be power line inspection, road inspection, offshore site monitoring, wind turbine inspections, and others. The utilization of cellular networks, such as the fifth-generation (5G) technology, is often considered to meet the requirement of wide-area connectivity. This study quantifies the performance of 5G-enabled UAVs when sensor data throughput requirements are within the 5G network’s capability and when throughput requirements significantly exceed the capability of the 5G network, respectively. Our experimental results show that in the first case, the 5G network maintains bounded latency, and the application behaves as expected. In the latter case, the overloading of the 5G network results in increased latency, dropped packets, and overall degradation of the application performance. Our findings show that offloading processes requiring moderate sensor data rates work well, while transmitting all the raw data generated by the UAV’s sensors is not possible. This study highlights and experimentally demonstrates the impact of critical parameters that affect real-life 5G-enabled UAVs that utilize the edge-offloading power of a 5G cellular network.