Tried rebuilding the document from not-really-scratch: I created a new document but I imported the text styles and I imported the text (from Word) without checking for errors there first. Same problem appeared a few pages into the document.
Next thing will be to remove all formatting from the Word text and see if it helps. Baah, I don't have time for this.
Some screen shots, with frame edges and non-printing characters showing, might help.
Yes, probably, but right now it seems like I've solved it.
Since everything was fine up to the point where I placed a new linked textbox on a specific page, I suspected that there might be something wrong with the text to be placed. So I deleted all text from a point slightly before that point and onward. Then I went back to my Word original, copied the text that I just deleted, pasted it (without formatting) into Text Edit, and then copied that text and pasted it into InDesign. After that, everything actually seems cool.
But probably I'll be back tomorrow saying that it didn't help. Let's see.
Can confirm that Word crap format info was the culprit. Note that I placed the text as "unformatted", but still, some weird info managed to sneak through. The trick to paste the text into a third app and then copy it from there worked fine.