Smartphones are today affordable devices, capable of embedding a large variety of sensors such as magnetometers or orientation sensors, but also the hardware needed to connect them to most wireless communication technologies such as Wi-Fi, Bluetooth, or cellular networks. Therefore, they are handy devices able to perform Received Signal Strength Indicator (RSSI) measurements for a wide variety of applications such as cellular coverage maps, indoor localization, or proximity tracking. However, to the best of our knowledge, the accuracy of such measurements has never been rigorously assessed. The goal of this paper is to assess the accuracy of the RSSI measurements made with a Commercial Off-The-Shelf (COTS) smartphone in a variety of conditions, and how possible inaccuracies can be corrected. We primarily focus on the LTE RSSI, but we also extend our results to the Bluetooth RSSI. In this paper, we build a controlled experimental setup based on commodity hardware and on open-source software. We evaluate the granularity and limitations of the Android API that returns the RSSI. We explore how reliable the measurements in a controlled environment with a monopolarized antenna are. We show that the orientation of the smartphone, the position or orientation of the source, and the transmission power have a significant impact on the accuracy of the measurements. We introduce several correction techniques based on radiation matrix manipulations and on machine learning in order to improve measurement accuracy to less than 5 dBm RMSE, as compared to a professional equipment. We also explore the reliability of measurements made in an outdoor realistic environment. We show that whereas transmission diversity available in LTE base stations significantly improves the measured RSSI regardless of the smartphone orientation, the Bluetooth RSSI remains largely sensitive to the smartphone orientation.