Migratory birds use different global cues including celestial and magnetic information to determine and maintain their seasonally appropriate migratory direction. A hierarchy among different compass systems in songbird migrants is still a matter for discussion owing to highly variable and apparently contradictory results obtained in various experimental studies. How birds decide whether and how they should calibrate their compasses before departure remains unclear. A recent ‘extended unified theory’ suggested that access to both a view of the sky near the horizon and stars during the cue-conflict exposure might be crucial for the results of cue-conflict experiments. In this study, we performed cue-conflict experiments in three European songbird species with different migratory strategies (garden warbler, Sylvia borin; pied flycatcher, Ficedula hypoleuca; and European robin, Erithacus rubecula; juveniles and adults; spring and autumn migrations) using a uniform experimental protocol. We exposed birds to the natural celestial cues in a shifted (120 deg clockwise/counterclockwise) magnetic field from sunset to the end of the nautical twilight and tested them in orientation cages immediately after cue-conflict treatments. None of the species (apart from adult robins) showed any sign of calibration even if they had access to a view of the sky and local surroundings near the horizon and stars during cue-conflict treatments. Based on results of our experiments and data from previous contradictory studies, we suggest that no uniform theory can explain why birds calibrate or do not calibrate their compass systems. Each species (and possibly even different populations) may choose its calibration strategy differently.