Abstract A recent stacking analysis of Planck HFI data of galaxy clusters led to the derivation of the cluster temperatures using the relativistic corrections to the Sunyaev–Zel'dovich effect (SZE). However, the temperatures of high-temperature clusters, as derived from this analysis, were basically higher than the temperatures derived from X-ray measurements, at a moderate statistical significance of 1.5σ. This discrepancy has been attributed by Hurier to calibration issues. In this paper, we discuss an alternative explanation for this discrepancy in terms of a non-thermal SZE astrophysical component. We find that this explanation can work if non-thermal electrons in galaxy clusters have a low minimum momentum (p 1 ∼ 0.5–1), and if their pressure is of the order of 20–30 per cent of the thermal gas pressure. Both these conditions are hard to obtain if the non-thermal electrons are mixed with the hot gas in the intracluster medium, but can be possibly obtained if the non-thermal electrons are mainly confined in bubbles with a high amount of non-thermal plasma and a low amount of thermal plasma, or are in giant radio lobes/relics in the outskirts of the clusters. To derive more precise results on the properties of the non-thermal electrons in clusters, and in view of more solid detections of a discrepancy between X-ray- and SZE-derived cluster temperatures that cannot be explained in other ways, it would be necessary to reproduce the full analysis done by Hurier by systematically adding the non-thermal component of the SZE.