得ない【えない】

得ない goes after a verb's ます-stem and means "can't possibly": the thing cannot happen or cannot be the case. It is read えない, and it belongs to writing and formal speech – with one exception, ありえないあり得ない, which you hear everywhere.

The positive うる得る reads える or うる, but the negative only ever reads えない: ×うない.

得ない or できない

得ない says whether something is possible at all. What a person is able to do, or allowed to do, takes the potential form or できない.

• × 私は泳ぎ得ない → 私は泳げない

In conversation, apart from あり得ない, the everyday ways to say "can't possibly" are the potential negative and はずがない.

あり得ない

From ある, あり得ない means "impossible", and in casual speech it is an exclamation: "No way!" It is often written in kana, ありえない.