The UK Research Excellence Framework (REF) introduced impact as an indicator in the evaluation of higher education research quality in 2014. Impact case studies (ICS) are the basis of this evaluation. ICS use "narrative" explanations of how research causes 'benefits to society'. This article analyses the evolving roles of the ICS and the relationships of ICS with various stakeholders. It draws on insights from legitimacy theory, Porter's analysis of the social meaning of objectivity, and the insight from economist Alan Goodhart on the dysfunctional transformation of indicators into targets. It highlights the challenges to ICS evaluators and shows how a spiral of mistrust can both undermine but also cause demand for numbers. It explores whether impact narratives represent an escape from dominant research evaluation metrics and concludes that they do not.