•Slovenian consumers in general are moderately doubtful of products bearing NHCs.•The trust in the formal authorities and social system are important for accepting foods with NHC/images.•Consumers who did not trust the social system also did not accept foods with NHC/images.•There are four groups of consumers regarding the reasons for their (non)acceptance of NHCs; To understand consumers’ acceptance and preference in relation to Nutrition and Health Claims (NHCs) on food products and the reasons for their (non)acceptance in an Eastern European country with a weak tradition in NHC, and an enormous increase in the number of foods with NHCs in recent years, this study aimed to fill a research gap by applying a multi-methodological approach combining a survey and conjoint analysis (n = 204) and a focus group (n = 45). The survey demonstrated that Slovenian consumers in general are moderately doubtful of NHCs. Conjoint analysis showed that when Slovenians choose their breakfast cereals, NHCs are more important than whether visual images are present. Visual images convince only the youngest consumers. The study revealed the importance of social perception and trust in the social system for accepting foods with NHCs/images. Consumers who did not trust the political and economic system also did not accept foods with NHCs/images.