Background/Objectives: Understanding people’s experiences is essential for improving healthcare systems (HCSs) and health outcomes, but assessing these experiences requires accurate, culturally and contextually appropriate measures. The People’s Voice Survey (PVS) is an instrument developed to assess current and future user experience across all levels of HCS globally. This research presents a cognitive interview (CI) approach as one measure for adapting PVS questions for adult and adolescent populations in three European countries. Methods: Twenty-four semi-structured CIs were conducted in Latvia, fifteen in Switzerland, and four in Germany. Study participants were recruited via convenience sampling. Interviews were conducted across virtual, telephone, and face-to-face settings. The “text summary” approach and Tourangeau’s four-step question-and-answer model served as the theoretical framework for the study. Results: The majority of the core PVS questions tested functioned well. The most common issues identified were related to comprehension and the decision process in all country cases. Analysis showed that almost all PVS core questions were also suitable for adolescents, with the exception of some questions related to evaluation and knowledge of HCS functioning, due to limited knowledge and/or experience with HCSs. Key adjustments included clarifying the question formulation, adding explanations to unclear and unknown terms and concepts, adding the following questions and missing answer categories. Conclusions: The paper shows how CIs are used to identify issues at all response stages when answering PVS core questions, thereby enabling the measure to be improved and supporting the use of CIs to enhance HCS assessment instruments.