V + 을 뻔했다 is used to show that something almost or nearly happened, usually as an unintended or unexpected result. Note that this is not to be confused with 거의:
직행 비행기를 타면 론돈에서 서울까지 거의 11시간 걸려요.
If you fly direct, it takes almost (practically, mostly, as good as) 11 hours to fly from London to Seoul.
늦잠을 자서 오늘 아침에 버스를 놓칠 뻔했어요.
I almost missed the bus this morning because I overslept.
모자를 잃어버릴 뻔했어요.
I almost forgot my hat!
자전거로 언덕을 내려가고 있을 때 너무 빨라서 넘어질 뻔했어요.
When I was going down the hill on my bike, I was travelling so fast that I almost fell off.
작년에 교통 사고로 죽을 뻔했어요.
Last year I almost died in a traffic accident.
졸업식을 위해 양복을 입은 친구들을 못 알아볼 뻔했어요.
When I saw my students wearing suits I almost didn’t recognise them.
한국어 배우는 걸 그만둘 뻔했어요.
I almost gave up learning Korean.
지난 번에 남자친구와 말싸움을 했을 때 헤어질 뻔했어요.
The last time I had an argument with my boyfriend we almost split up.