School dreams are among the most common. In a survey of 1,181 Canadian university students, 67.1 % said they had ever dreamed of school, teachers or studying, and 45.0 % of failing an exam. The youngest students, 18 and under, reported exam-failure dreams most often.
In the Typical Dreams Questionnaire study (Nielsen et al., 2003), "school, teachers, studying" was the fourth most common theme (67.1 %; women 71.3 %, men 56.9 %). "Failing an examination" ranked eleventh at 45.0 %: 51.1 % of students 18 and under and 41.7 % of those 19 and over.
These were students, so the study says nothing about adults who still dream of exams long after school.
What the theories say
The continuity hypothesis proposes that dreams draw on waking life (Schredl & Hofmann, 2003). For a group of students, school is waking life. It is a theory, not proof of why any one person dreams of exams.
Questions
Why do I dream about failing an exam?
It's common: almost half of students in a large survey had had the dream. One theory, the continuity hypothesis, says dreams draw on waking life; that's a theory, not a fact.
Is it normal to dream about school?
Yes. About two in three students in a large survey said they had ever dreamed of school, teachers or studying.
Sources
Nielsen, T. A., Zadra, A. L., Simard, V., Saucier, S., Stenstrom, P., Smith, C., & Kuiken, D. (2003). The typical dreams of Canadian university students. Dreaming, 13(4), 211–235. https://doi.org/10.1023/B:DREM.0000003144.40929.0b
Schredl, M., & Hofmann, F. (2003). Continuity between waking activities and dream activities. Consciousness and Cognition, 12(2), 298–308. https://doi.org/10.1016/S1053-8100(02)00072-7 (theory beat only)
73.8 %of students said they'd dreamed of falling
81.5 %said they'd been chased: the top dream of 55
18.8 %said they'd dreamed of losing their teeth