To say that something bad almost happened (but didn't), you can add the word 差点 (chàdiǎn) before the verb. You will also hear 差点儿 (chàdiǎnr) in northern China. There is no difference in meaning between 差点 and 差点儿.
The word 差 (chà) has a lot of meanings. In this case, it means "to lack," or "to be short." So in Chinese, the way to say "almost" is to say, "lacking that little bit." If that "little bit" hadn't been lacking, it would have happened. But it was lacking, so it didn't happen.
"差"这个字有很多意思。单字的意思是"to lack"或"to be short"。因此，在中文里，表达"almost"的方式就是说"lacking that little bit"，要是"那一点（little bit）"存在的话，那事情本会发生，但因为"那一点"不存在，所以没发生。
Subject + 差点(儿) + [Verb Phrase] + 了
Note that the "Verb Phrase" part of the structure is usually something bad or something you don't wish for, and that it didn't happen. You can put "就" after "差点儿" .
我们队差点儿（就）输了。 (the team didn't lose)
wǒmen duì chàdiǎn er jiù shū le.
Our team almost lost.
雨很大，我差点（就）回不来了。 (I actually did make it back, though)
yǔ hěn dà, wǒ chàdiǎn jiù huí bù lái le.
The rain is so heavy I almost couldn't make it back.
路上堵车，我差点儿（就）迟到了。 (I wasn't late)
Lùshàng dǔchē, wǒ chàdiǎn er jiù chídào le.
The traffic is bad; I was almost late.
我差点儿（就）犯了一个大错。 (I didn't make a big mistake)
wǒ chàdiǎn er jiù fàn le yīgè dà cuò.
I nearly made a big mistake.
汽车差点儿（就）到机场了，可是突然坏了。 (the bus didn't arrive)
qìchē chàdiǎn er jiù dào jīchǎng le, kěshì túrán huài le.
The bus almost arrived at the airport.It suddenly broke down.
Logically, 差点没 is not that different from 差点. In fact, you shouldn't think of it as a single word; think of the 没 as belonging to what follows 差点. Still, it can be a little confusing, and is worth a closer look.
Subject + 差点 + 没 [Verb Phrase]
Note that the "Verb Phrase" part is a desired result. Also note that because you're using 没, you don't use 了 after the verb phrase.
我差点没赶上火车。 (I almost missed it, but I didn't.)
wǒ chà diǎn méi gǎn shàng huǒ chē.
I almost didn't make the train.
我儿子差点儿没考上大学。 (He just barely passed the exam.)
wǒ ér zǐ chà diǎn ér méi kǎo shàng dà xué.
My son almost failed the college entrance examination.
时间太紧，这个项目差点儿没做完。 (We almost didn't finish, but we did.)
shí jiān tài jǐn, zhè gè xiàng mù chà diǎn ér méi zuò wán.
Time was almost up, we almost didn't finish this project.
这次对手太强了，我们队差点儿没赢。(But we won.)
zhè cì duì shǒu tài qiáng le, wǒ men duì chà diǎn ér méi yíng.
This time, the opponent is so strong that our team almost didn't win.
排队的人太多了，差点儿没买到。(But I did get it.)
pái duì de rén tài duō le, chà diǎn ér méi mǎi dào.
The waiting line is too long. I almost didn't get it.
Translated from: allsetlearning
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