The narrator makes a segue from the princes' education towards the king's status, and describes that one night, Don Fernando had a bad dream.
In that dream, Don Juan was attacked and killed by two people and then left to rot in a deep well. It disturbs the king so much that he soon falls ill, and many healers are called in from all over.
Most of them weren't able to diagnose what was wrong with the king, until this one man comes and says that Don Fernando's illness is from a bad dream. He then says that to cure it, he only needs to listen to a song from a bird called the Adarna, that lives in the Piedras Platas tree located in the Tabor mountains.
The healer then describes some of the bird's habits, particularly how it goes off to feed somewhere else in the daytime and comes back to its perch at night. And he also adds that they would only need to catch it and bring it to the kingdom so its song would surely heal the king.
And so, encouraged by this, Don Fernando calls on his eldest son, Don Pedro, to have him go out and retrieve the bird.
This reminds me of some other real-life stories of people declining in health simply because someone they really loved passed away or left them in some other way, like a breakup or something. And for some of these people, even the very thought of having these loved ones disappear from their lives just takes a toll on them.
So, to a degree, I'm not too surprised that someone could fall ill just because they dreamt of someone they knew and cared for getting killed. But for me, it would have to be the sort of someone who wasn't strong mentally and emotionally.
Also, another point I've been thinking about regarding this scene is how the king couldn't tell people about the dream or something like that. Like, was he scared of having the dream come true if he told people about it? Or did he tell about it, but the other people who came to try and cure him didn't think it was a factor?
But either way, he did fall ill, and the only hope of a cure they got was that thing about the Adarna bird.