Verb + -(으)ㄹ 뻔했어요 

It means ‘almost happened’; we use this expression for an event that you don’t wish to happen.  The infinitive form is ‘-(으)ㄹ 뻔하다‘, however we always use this 뻔했어요 which is the past tense; 뻔해요(X), 뻔할 거예요(X).  We often use this with ‘하마터면(adverb)‘ which means ‘almost’.

For example,
넘어지다 (fall down)
길이 너무 미끄러워서 넘어질 뻔했어요.  Because the road was too slippery, I almost fell down.
스케이트를 타다가 넘어질 뻔했어요. While I was skating, I almost fell.
실수하다 (make mistake)
요즘 정신이 없어서 큰 실수할 뻔했어요.  Because I was too busy lately, I almost made a big mistake.
큰일 나다 (get into trouble)
큰일 날 뻔했어요. almost got into trouble
다치다 (hurt)
하마터면 크게 다칠 뻔했어요. almost hurt badly.
잊어버리다 (forget)
요즘 너무 바빠서 하마터면 중요한 약속을 잊어버릴 뻔했어요. I have been too busy recently, so I almost forgot an important appointment.
죽다 (die) * We sometimes say ‘죽을 뻔했어요’ to emphasize an event.
차 사고가 나서 죽을 뻔했어요. Because of that car accident, I almost died.
물에 빠져서 죽을 뻔했어요. Because I fell into water, I almost died.
너무 추워서 죽을 뻔했어. It was too cold, I almost died!
목이 말라서 죽을 뻔했어요. I was too thirsty, I almost died!
무서워서 죽을 뻔했어요. I was too scared, I almost died!