The first failure point is a scripted self-introduction. Reciting a prepared text sounds polished, but it breaks the moment the interviewer asks a follow-up, because what comes next is not in your own words. A steadier method is to prepare a few anchor points rather than a script: three or four keywords that hold the content you want to convey. The wording can differ each time; the substance stays the same. That way, an interruption does not derail you.
The second failure point is freezing on a follow-up question. Many candidates read a follow-up as meaning they got it wrong; more often the interviewer simply wants to see whether you can elaborate. When you genuinely do not know, saying so — and describing how you would go about finding out — beats silence or improvising. An invented answer invites a second follow-up, and that is where it becomes hard to recover.
The third failure point is knowing nothing specific about the school. Statements like 'your school has a long history and a good ethos' apply to nearly every school, which makes them empty. Before the interview, find at least one concrete thing you are actually interested in — how a particular subject is structured, an activity you looked up, something published by the school — and explain why it connects to you. It takes half an hour of research and it shows.
Note that arrangements, quotas and formats differ between schools and change from year to year, and some schools do not offer a knocking-door stage at all. The above covers general preparation only; always check what the school has published for the current year.