There is now a limit of 20 repeats on the quiz-section of the Settler application. After 10 tries the page will notify the user that they will get banned if they try too often, without saying when the ban will actually take place. They are then barred from taking the test again and from entering the server.
I had now a counter running for some time to see how often people fail a certain section of the application process and we had now 100 people start the quiz. 23 of the 100 needed more than the limit and 5 did more than 50 tries. Around 10 tries is the average number of tries people do, but heavily outweighed by the people who do so many tries.
So I think to ban someone after 20 tries is justified, since they obviously did not read the website but just try to click random stuff and brute-force themselves through the application.