Reducing rates of degree non-completion and widening participation in STEM (Science, Technology, Engineering and Mathematics) are key international policy concerns. This paper analyzes open-ended response survey data from 1886 degree students in England, focusing on the 27% (n = 136/501) STEM and high-status medicine (STEMM) degree students and 26% (n = 367/1385) non-STEMM students who expressed concerns that they might not complete their degree. Insights are supplemented with two illustrative longitudinal case studies of young women, interviewed from age 10 to 21, who withdrew from their respective STEM degree courses. Around a quarter of both STEMM and non-STEMM students expressed concerns that they might not complete and reported similar types of concern, with academic concerns being of prime importance--but particularly so among STEMM students. The percentage of students expressing concerns varied between STEMM disciplines (from 18% of maths students to 37% of computing students) and by student demographics, with women, racially minoritized, and students from less affluent backgrounds appearing most likely to worry that they might not complete. A Bourdieusian lens is employed to interpret patterns in the findings, challenging human capital explanations of non-completion as due to individual student deficit ("misfits") with a sociostructural interpretation ("misrecognition") that draws attention to the role of capital and symbolic violence in shaping differential risks of non-completion. Calls are made for the value of understanding STEMM student non-completion as a structural issue, not just individual "unprepared" students who do not "fit" their degree.