I'm not talking here about the stress exams put students under, although I could. I'm talking here about how exams are one of the worst possible ways of testing whether someone is a good mathematician.
Here's the problem: maths exams are tests of a) memory and b) calculation. You can do well in almost any maths exam by getting hold of all of the past papers and working through them. That's what I spend about 80% of my time as a maths tutor doing, because it's the most effective method I know for getting students a good grade in the exam.
It doesn't need to be that way. The most important skills - in my opinion - for a mathematician to have are:
Virtually none of those come up in an exam. The last one, sure; possibly modelling real-life problems (although normally you're spoon-fed the model the examiners want) and possibly persistence (although you're limited by how much time has been arbitrarily allotted to the exam).
However looking things up and talking with your colleagues - two of the most important skills as a mathematician, and two of the most transferable mathematical skills - are considered 'cheating' and will lead to you failing the exam and probably having other qualifications revoked as well.
Let me say that again: the exam system actively penalises two of the most important skills that can be learned in maths.
Meanwhile, they encourage things like last-minute cramming, following instructions blindly, and teaching to the test - which I do, because my job is to ensure that little Jimmy gets the grade he needs to get to university. (I'll do my best to foster his curiosity and get him enthusiastic about maths, but if I do that and he still does poorly, it does nobody any good).
Exams are designed for convenience. An examiner can sit down and grade your work (you got 80% of the available marks, so you deserve to get a letter A! have a cookie) and a university can say 'you have the magic letters A, B and B, so you're allowed to come here and study for more exams in much the same vein', or so an employer can differentiate between candidates without having to bother interviewing them.
This isn't just silly, it's actively harmful. We end up with a workforce whose main characteristics are docility, a good short-term memory and the ability to work in silence under pressure.
And I don't think those are the traits we should be emphasising.