Organic field-effect transistors exhibit operational instabilities involving a shift of the threshold gate voltage when a gate bias is applied. For a constant gate bias the threshold voltage shifts toward the applied gate bias voltage, an effect known as the bias-stress effect. Here, we report on a detailed experimental and theoretical study of operational instabilities in p-type transistors with silicon-dioxide gate dielectric both for a constant as well as for a dynamic gate bias. We associate the instabilities with a reversible reaction in the organic semiconductor in which holes are converted into protons in the presence of water and a reversible migration of these protons into the gate dielectric. We show how redistribution of charge between holes in the semiconductor and protons in the gate dielectric can consistently explain the experimental observations. Furthermore, we show how a shorter period of application of a gate bias leads to a faster backward shift of the threshold voltage when the gate bias is removed. The proposed mechanism is consistent with the observed acceleration of the bias-stress effect with increasing humidity, increasing temperature, and increasing energy of the highest molecular orbital of the organic semiconductor.