You don't need to fast to completion for healing, and depending upon what it is, you may not even need to fast (though fasting is better at sorting out some things more than others).
I think the reason people get their illnesses back again is because they make no long term changes, and go back to the same old ways that got them into trouble before the fast.
Dr Joel Fuhrman got it right in his book Fasting and Eating for Health, when he said:
"If a person is unable or unwilling to make permanent changes in the diet and to follow a healthy eating plan before and after the fast, there really is no point in fasting."