Societal impact considerations play an increasingly important role in research evaluation. In particular, in the context of publicly funded research, proposal templates commonly include sections to outline strategies for achieving broader impact. Both the assessment of the strategies and the later evaluation of their success are associated with challenges in their own right. Ever since their introduction, altmetrics have been discussed as a remedy for assessing the societal impact of research output. On the basis of data from a research center in Switzerland, this study explores their potential for this purpose. The study is based on the papers (and the corresponding metrics) published by about 200 either accepted or rejected applicants for funding by the Competence Center Environment and Sustainability (CCES). The results of the study seem to indicate that altmetrics are not suitable for reflecting the societal impact of research that was considered: The metrics do not correlate with the ex ante considerations of an expert panel.