Can someone sue after they have settled with the insurance company in a no-fault state?
It depends on the terms of the settlement and also whether the settlement paid for all costs, losses, etc.
1) If the settlement agreement stated that the payment was in full satisfacation of all claims, then that is enforceable and there can be no further lawsuit. Similarly, an agreement to not sue is also enforceable.
2) If the injured party was paid everything due him or her (e.g. say there was some property damage and medical bills, but no pain and suffering, and the property and bills were completely paid for) there is no grounds for a suit, since the claim has been fully satisfied.
Only if there was no agreement that the settlement was full satisfaction, no agreement to not sue, and the settlement left some bills, costs, etc. unpaid could there be a lawsuit.