
durchfallen
to fail
To not pass an exam, test, or class.
Leider ist er bei der Fahrprüfung durchgefallen.
Unfortunately, he failed the driving test.
to fail / fall through
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to fail
To not pass an exam, test, or class.
Leider ist er bei der Fahrprüfung durchgefallen.
Unfortunately, he failed the driving test.

to fail
To not receive approval in a vote, evaluation, or from an audience and be rejected. Often refers to laws, proposals, or artistic performances.
Der Gesetzesentwurf ist im Parlament mit knapper Mehrheit durchgefallen.
The draft law failed in parliament by a narrow majority.

to fall through
To physically fall downward through an opening or an unstable surface.
Vorsicht, der alte Holzboden ist morsch, man könnte durchfallen!
Careful, the old wooden floor is rotten; someone could fall through!
'Durchfallen' (to fail / fall through) is a separable verb. In a main clause, the prefix 'durch' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er fällt oft bei Prüfungen durch.
He often fails exams.
The perfect tense of 'durchfallen' (to fail / fall through) is always formed with 'sein' (to be), not with 'haben' (to have). This applies to all meanings of the word.
Ich bin letztes Mal leider durch die praktische Prüfung gefallen.
Unfortunately, I failed the practical exam last time.
The literal meaning 'physisch durch etwas fallen' (to physically fall through something) helps you understand the figurative meanings. In an exam, you more or less 'fall' through the net of requirements.
Pass auf, der Stuhl ist kaputt, nicht dass du durchfällst!
Watch out, the chair is broken—don't fall through it!
When talking about an exam, you use the preposition 'bei' or 'in', followed by the dative. Both are correct.
Sie ist in der mündlichen Prüfung durchgefallen.
She failed the oral exam.
In the meaning 'abgelehnt werden' (to be rejected), the preposition 'bei' + dative is often used to name the group that rejects something.
Der Vorschlag ist bei der Abstimmung im Team durchgefallen.
The proposal failed in the team vote.
to reject
With a suggestion, ablehnen can refer to the active decision made by a person or institution. Durchfallen, by contrast, describes a suggestion failing to receive enough support in a vote or evaluation. Use ablehnen when the rejecting person or institution is important, and durchfallen when the lack of agreement is the focus.

Der Vorschlag fiel in der Abstimmung durch.
The proposal failed in the vote.
Der Ausschuss lehnte den Vorschlag in der Sitzung ab.
The committee rejected the proposal during the meeting.