Emme pystyneet avaamaan ovea.

In Finnish, negation is done with a special negative verb, and that negative verb changes according to the person.

So:

• en = I do not
• et = you do not
• ei = he/she/it does not
• emme = we do not
• ette = you plural do not
• eivät = they do not

So emme already includes the idea of we.
That is why Finnish does not need a separate subject pronoun here.

You can say Me emme pystyneet avaamaan ovea, but me is usually added only for emphasis, contrast, or clarity.

The past tense is shown by pystyneet.

In Finnish negative past sentences, the pattern is:

• the negative verb in the correct person/number
• plus the main verb in a special past form

Here:

• emme = we do not / we did not
• pystyneet = past form of pystyä

So emme pystyneet means we were not able / we could not.

Compare:

• emme pysty = we are not able
• emme pystyneet = we were not able

Pystyneet comes from the verb pystyä, which means to be able to, to manage to, or to be capable of.

In this sentence, pystyneet is the form used in a negative past construction.

You can compare:

• pystyn = I am able
• pystyimme = we were able
• emme pystyneet = we were not able

The ending -neet is the plural form used here because the subject is we.

Compare:

• en pystynyt = I could not
• emme pystyneet = we could not

Because the verb pystyä is followed by the third infinitive illative, which usually ends in -maan / -mään.

So:

• tehdä → tekemään
• mennä → menemään
• avata → avaamaan

With pystyä, Finnish says literally something like to be able into doing/opening, even though that sounds unnatural in English.

So:

• pystyä avaamaan = to be able to open
• emme pystyneet avaamaan = we were not able to open

This is a very common structure in Finnish, and it must simply be learned with pystyä.

Because Finnish normally uses the partitive object in negative sentences.

That is one of the most important object rules in Finnish.

So even if English says the door, Finnish uses partitive after negation:

• Avaamme oven. = We open / will open the door.
• Emme avaa ovea. = We do not open the door.
• Emme pystyneet avaamaan ovea. = We could not open the door.

So here ovea is partitive because the whole clause is negative.

Because ovi is one of those nouns whose stem changes in certain cases.

Its basic forms are:

• ovi = door
• oven = door's / the door (genitive)
• ovea = door (partitive)
• oveen = into the door
• ovessa = in the door

So the stem often becomes ove- in oblique cases.

This is just part of how the noun ovi is inflected.
So in this sentence:

• dictionary form: ovi
• partitive singular: ovea

Yes, absolutely.

Both are correct:

• Emme pystyneet avaamaan ovea.
• Me emme pystyneet avaamaan ovea.

The version without me is more neutral and natural in many contexts, because emme already tells you the subject is we.

Adding me gives extra emphasis, for example:

• Me emme pystyneet avaamaan ovea, mutta he pystyivät.
  We couldn’t open the door, but they could.

They are similar, but not always identical.

• voida often means can / may / be possible
• pystyä often emphasizes ability, capacity, or managing to do something

So:

• Emme voineet avata ovea can mean we could not open the door
• Emme pystyneet avaamaan ovea also means we could not open the door, but it often feels more like we were unable / we didn’t manage

There is also a grammar difference:

• voida is usually followed by the basic infinitive: voida avata
• pystyä is followed by the -maan/-mään form: pystyä avaamaan

That sentence is also correct, but it means something slightly different.

• Emme avanneet ovea = We did not open the door.
• Emme pystyneet avaamaan ovea = We could not open the door / We were unable to open the door.

The version with pystyä adds the idea of inability, not just the fact that the action did not happen.

So the original sentence is specifically about not being able to do it.