The A.Word.A.Day mailing list expands on Webster 1913's definition:

aegrotat (EE-gro-tat, ee-GRO-tat) noun: An unclassified degree granted a university student who has fulfilled all requirements for graduation but was prevented by illness from attending the final examinations.

From Latin aegrotat literally, he is sick, equivalent to aegrot(us) sick (aeg(e)r sick + -otus adjectival suffix) + -at 3rd singular ending.

"Powell got a first in the exams he sat in the nursing home - his classics tripos finals. If he had not been able to sit them there, he would only have got an aegrotat degree - which might well have affected his future."
Eric James, "Deep rivers, indeed," New Statesman, Apr 24, 1998.