The thermal smearing of the positron-annihilation-angular-correlation curves is studied. In the case where the medium is a homogeneous interacting electron gas in thermal equilibrium, the analysis applied by Stewart and co-workers to the thermal smearing is shown to correctly give the effective mass. In conjunction with previous calculations of the effective mass, our results indicate that the observed thermal smearing cannot be explained as mainly caused by electron-gas effects. It is shown how the model can be modified to take into account the positron-phonon interaction. This reduces the discrepancy between theory and experiment, but not sufficiently to account fully for the experimental data. It is argued that new experiments are needed before one can decide whether it is necessary to consider new mechanisms in order to explain the effect. A convenient formalism is presented, which allows one to calculate all positron quantities at moderate nonzero temperatures within a given approximation scheme. The "ladder" approximation is used to illustrate the argument and to estimate correction terms.