Yes, there are people who have trouble with public speaking to a debilitating degree, but it would be excessively rare for someone to not at the very least in a one on one with their professor/teacher be able to be so badly affected as to not seem they've even heard of a topic or at least be able to prove they've worked on it to a certain degree.

I would be immediately skeptical of any student who claims they are completely unable to explain their knowledge unless they are allowed to work in complete isolation with nobody to monitor they aren't cheating in some way.