Peer review has developed over time to become the established procedure for assessing and assuring the scientific quality of research. Nevertheless, the procedure has also been variously criticized as conservative, biased, and unfair, among other things. Do scientists regard all these flaws as equally problematic? Do they have the same opinions on which problems are so serious that other selection procedures ought to be considered? The answers to these questions hints at what should be modified in peer review processes as a priority objective. The authors of this paper use survey data to examine how members of the scientific community weight different shortcomings of peer review processes. Which of those processes' problems do they consider less relevant? Which problems, on the other hand, do they judge to be beyond remedy? Our investigation shows that certain defects of peer review processes are indeed deemed irreparable: (1) legitimate quandaries in the process of fine-tuning the choice between equally eligible research proposals and in the selection of daring ideas; and (2) illegitimate problems due to networks. Science-policy measures to improve peer review processes should therefore clarify the distinction between field-specific remediable and irremediable flaws than is currently the case.