It was a very touching story and, while I don't presume to speak for the Duggars at all, it very well might explain why they don't practice any form of planning other than God's will. Who are we to judge?
Not judging, just observing that the pace of her pregnancies might be taking it’s toll. Her last several pregnancies were by c section and the uterus can take being cut open only so many times. Nutrients are depleted, the uterus needs time to heal and rebuild before being used again.
There is nothing wrong in God’s eyes to practice NFP and abstain for several days per month. We are human beings and we have self control. We bash single mothers and fathers and married people who have more kids than they can care for but we celebrate these people just because they’re not on welfare. But they are all doing the same thing—leaving the results of their sexuality to God’s will. Yes it is God’s will even if a baby is born to a crack whore!!
It would normally be their business and no one else’s but they have not remained private. They put their lives on television for the world to see. Therefore we can make observations about them.