Nem tudok egyszerre írni és telefonálni.

Yes, structurally it is very close to I not can.

• nem = not
• tudok = I can / I am able to

Hungarian usually makes a sentence negative by putting nem directly before the verb:

• Tudok írni. = I can write.
• Nem tudok írni. = I cannot write.

So nem tudok is the normal way to say I can’t.

Because -ok shows the subject I.

The verb is tud = to know / to be able to, and here it is conjugated in the first person singular:

• tudok = I can
• tudsz = you can
• tud = he/she/it can

Hungarian verbs usually show the subject in the verb ending, so the pronoun én (I) is not needed unless you want extra emphasis.

That is a very common question. tud can express both knowing and being able to, depending on the structure.

• Tudok valamit. = I know something.
• Tudok írni. = I can write.

When tud is followed by an infinitive, like írni or telefonálni, it usually means can / be able to.

So in this sentence, tudok írni és telefonálni means I can write and talk on the phone.

Because after tud, Hungarian normally uses the infinitive.

The -ni form is the infinitive, like English to write, to call, to speak.

• írni = to write
• telefonálni = to telephone / to talk on the phone

So:

• tudok írni = I can write
• tudok telefonálni = I can talk on the phone

This is very similar to English can + verb, except Hungarian uses the infinitive form explicitly.

Because Hungarian puts the idea of to inside the infinitive ending -ni.

So instead of having a separate word, Hungarian uses:

• írni = to write
• telefonálni = to telephone

That means -ni already does the job that English to does.

Egyszerre means at the same time, simultaneously, or all at once, depending on context.

In this sentence, it means at the same time:

• egyszerre írni és telefonálni = to write and talk on the phone at the same time

A useful thing to remember is that egyszerre often refers to two actions happening simultaneously.

It is placed there because it modifies the whole action write and talk on the phone at the same time.

Hungarian word order is flexible, but position often shows focus or what belongs together. Here, egyszerre naturally introduces the idea of simultaneity before the two coordinated infinitives:

• Nem tudok egyszerre írni és telefonálni.

This is a very natural word order.

You can also say:

• Nem tudok írni és telefonálni egyszerre.

That is also correct, but it may sound slightly different in emphasis. Both mean the same basic thing.

Because Hungarian, like English, can join two infinitives with a single and:

• írni és telefonálni = to write and to talk on the phone

Hungarian does not need to repeat tudok or add anything extra:

• Nem tudok írni és telefonálni.
• not: Nem tudok írni és tudok telefonálni.

The two infinitives are simply coordinated under one verb, tudok.

Yes:

• Én nem tudok egyszerre írni és telefonálni.

This is grammatically correct, but én is usually added only for emphasis or contrast, such as:

• Én nem tudok egyszerre írni és telefonálni, de ő igen.
  • I can’t write and talk on the phone at the same time, but she can.

Without special emphasis, Hungarian usually leaves the pronoun out because tudok already shows I.

Yes, telefonálni is a normal and natural verb meaning to phone / to talk on the phone.

It comes from:

• telefonál = he/she phones
• telefonálni = to phone

So the sentence sounds natural and everyday. A learner should be comfortable treating telefonálni as a standard verb.

Not really in normal usage. In this structure, nem tudok + infinitive is usually understood as I can’t / I’m not able to.

So:

• Nem tudok egyszerre írni és telefonálni.

is understood as I can’t write and talk on the phone at the same time, not as I don’t know how to.

Hungarian tud can relate to both knowledge and ability, but with action verbs in this kind of sentence, ability is the natural interpretation.