I, too have noticed this. It is the design of the web page. I've noticed that, sometimes, when a web page has more than one page (as in, you have to click "next page"), the Reader function will pick up the whole article. And then, sometimes, it won't. But, that's not the Reader fault - its the way the page has been designed.
So, the only thing to do is to finish with the first part, close the Reader, and go to the next part.
Sorry to not be of more help.
Marilyn