Not only that, it's not even to do with "boredom" in any sense that I understand the word:
"participants limited TV, computer and video games, music, and telephone time to 30 min per day; in the time made free thereby, they were to read, sit quietly, write in their journals, meet with friends, and so on"
Calling that "boredom", says more about whoever wrote the title "Why Kids Need To Be Bored" than it says about kids' performance and boredom.
Then, the result of this study seems to be: out of 3 participants, 1 showed improvement, the other 2 dropped out. No data on 2 out of 3 test subjects, really, statistically proves nothing at all. It could go either way, or completely different. And it's bad science to not clearly point this out in the abstract. There's no shame in a failed experiment, but there is a lot of shame in pretending that it shows a trend that you didn't actually measure.
I'd be very surprised if research points out that more boring study material leads to improved results. Actually, with N=1 I won't be surprised no matter what.