One of the reviews of The Zero Blessing asked, quite reasonably, how Caitlyn managed to enter Jude’s in the first place. Why wasn’t there an entrance exam? Or even a test for magical ability?
The short answer is that there is no entrance exam. Anyone willing to pay the fees (or have their parents pay the fees) can get a place at Jude’s. The general assumption is that anyone can use magic, so it’s really just a matter of practicing and practicing until even the slowest students can use spells. (Or keep failing the exams to move up to the next level, at the end of the first year.) Cat’s father registered her (and her sisters) and no one thought anything of it. They would have been more suspicious if he hadn’t.
Rose was tested for magical ability – it was how she won the scholarship. (If you consider magic to be similar to music, Rose is the untrained talent who could be great with some proper tutoring.) It was assumed that she would pick up all the background knowledge from the other students, which is – to be fair – what happened.
Obviously, there are details I’m not going to go into about how magic works in this world, because they’re major plot points for the next two books. However, if you think about it, you’ll see that Cat’s is actually in a very lucky position – she has both the background knowledge and the training she needs to make use of her true talents. Others like her might well pass unnoticed.