We report on the results of a comprehensive study of THz emission from a set of dedicated AlGaN/GaN high-electron-mobility transistors. We find that voltage-biased transistors indeed emit in the THz frequency range, as reported in the literature; however, our data let us conclude that this radiation cannot be directly attributed to plasmonic instability phenomena. Instead, two other distinct mechanisms are identified. One is based on high-frequency self-oscillations originating from positive feedback within the frequency range where the transistor provides gain. Such oscillations are especially facilitated by the integration of antenna structures and cease to exist after taking specific measures for circuit stabilization. Another mechanism is identified for the case of broadband emission from multifinger transistors fabricated without any specific antenna. In contrast to the predictions of the plasmonic instability picture, the spectra of this emission depend on the gate and drain biasing conditions and on the bias modulation frequency. This emission can be understood as a combination of thermal emission from the heated material and from thermally excited plasmons and trap states.