Doktor kaže da ne smijem opet ozlijediti rame.

Croatian often drops subject pronouns because the verb ending already tells you who the subject is.
Here, smijem is the 1st person singular form, so it already means I am allowed / I may.

You could say ja ne smijem, but ja would add emphasis, as if contrasting with someone else.

Da introduces a content clause, similar to English that.

So Doktor kaže da... means The doctor says that...

In Croatian, this is the normal way to introduce what someone says, thinks, knows, believes, and so on.

Ne smijem is from the verb smjeti, which is about permission or prohibition.

In this sentence, ne smijem means something like:

• I must not
• I am not allowed to
• I shouldn't (depending on context)

It is stronger than simple inability.
So it is not the same as ne mogu, which means I can't / I am unable to.

Because smjeti is a modal verb, and modal verbs are commonly followed by an infinitive in Croatian.

So:

• smijem doći = I may come
• ne smijem trčati = I must not run
• ne smijem ozlijediti = I must not injure

This works a lot like English modal structures, except Croatian uses a plain infinitive without any word like to.

This is a question of verbal aspect.

• ozlijediti is perfective
• ozljeđivati is imperfective

The perfective ozlijediti presents the injury as a single complete event: to injure the shoulder again one more time.

That fits this sentence well, because the doctor is warning against another whole injury event.

If you used ozljeđivati, it would sound more like repeated or ongoing injuring in a general sense.

Because here rame is the direct object.

• ozlijediti rame = to injure a shoulder
• ozlijediti se = to get injured / injure oneself

So the sentence is built as injure the shoulder, not injure oneself.

If you said ozlijediti se, the focus would be on the person getting injured, without naming the shoulder as the direct object in the same way.

Rame is in the accusative case because it is the direct object of ozlijediti.

The reason it looks unchanged is that rame is a neuter noun, and for many neuter nouns, the nominative singular and accusative singular have the same form.

So:

• nominative: rame
• accusative: rame

Even though the form is the same, its function in the sentence is accusative.

Croatian often leaves possession unstated when it is obvious, especially with body parts.

So ozlijediti rame naturally means injure my/the shoulder, depending on context, and here it is clearly understood to be the speaker's shoulder.

If you want to make the possession explicit, you could say something like svoje rame, but that is often unnecessary.

Opet means again.

In this sentence, it modifies ozlijediti rame, so the meaning is injure the shoulder again.

Its position is natural and common:
ne smijem opet ozlijediti rame

Croatian word order is fairly flexible, but different placements can change emphasis slightly. Putting opet before the infinitive makes it very clear that the repeated action is the injury.

Both are used, but they are not identical in tone.

• liječnik is the standard, specifically medical word for physician/doctor
• doktor is very common in everyday speech and is perfectly natural here

One thing to remember is that doktor can also mean someone with a doctorate, depending on context.
But in a sentence about medical advice, everyone will understand it as doctor in the medical sense.

Kaže is the present-tense form of kazati or reći used in a present-time reporting frame: The doctor says...

Croatian often uses the present here just as English does.

If you wanted to report something the doctor said earlier, you would more likely use a past form such as rekao je. So the tense depends on whether the speaking is presented as current or past.