It's an interesting and valid question.
As joe says, ILR refusals are not common, but they can and do happen.
Don't be alarmed though, all of the ones I am aware of have involved very complicated situations that should, in my opinion, have been reviewed by an immigration advisor.
Provided the ILR application meets all requirements then it will be approved.
In the case you mention regarding Children. If for some reason the ILR could not be approved, and the application mentioned the children, then in my opinion the ILR would still be refused.
The ILR cannot be approved if it is not compliant.
Apart from the appeals system there is also something called Discretionary Leave to Remain.
You find read more about Discretionary Leave to Remain here
Look to the bottom of the page
I know of a number of cases where this was applied.
I'm not saying this is exactly what will happen. Just that both appeals and DLR are recognised routes if ILR is refused.
More research for you lastlid.
