The relationships between the internationalisation of higher education and language are still poorly understood. We foreground the perspective of students in order to advance our understanding of these interrelations in the context of the consolidation of the European Higher Education Area (EHEA). Accordingly, we propose gathering answers to the questions: "What, from students' perspectives, are their experiences and perceptions of pluri- and multilingualism in the context of the internationalisation of higher education?" "And how are they dealing with these experiences and perceptions?" Existing studies confirm that local specificities pertaining to languages and education systems impact substantially on the answers to these emerging questions. Thus, an overall picture of the interactions between language and internationalisation that shape the EHEA require the integration of findings from localised investigations that bring to light these particularities. Here we present an overview of findings from a mixed methods study in a medium-sized, historically German-language university in multilingual Switzerland. Our findings confirm that students are making considerable efforts to ensure that their plurilingualism extends beyond English. Notwithstanding attempts by the university to respond to their needs, students however still struggle with the tensions between what they can actually do, what they report they would like to do, and what they perceive is expected of them concerning language competencies during their studies and after. On the basis of these reflections, lecturers at the Language Centre developed a four-language course "Communication training in multilingual settings" that uses French, Italian, English, and German. This course is briefly introduced as an example of a language training intervention which seeks to move beyond the persistent constraints of a compartmentalised approach to pluri- and multilingualism in higher education.