Clearly if you fall pregnant you have to deliver the baby somehow so doctors can't refuse to perform a c-section no matter how many you've had previously.
Attempting a VBAC after your third c-section is considered v dangerous due to the large amount of scar tissue, weakened areas of the uterus, increased potential for the uterus to rupture etc.
I know that most (if not all) doctors will strongly advise you to be sterilized after your third section due to the massively increased risks to your health if you accidentally fell pregnant again.
From the article it seems that this woman has ignored any and all medical advice she has been given but still thinks the NHS should have magically prevented anything from going wrong.
It is an awful thing to have happened to her but I fail to see how she could have had so many c-sections without being aware of the risks.
It was her decision to have the number of children she does and gamble with her health the way she has.