North Carolina law treats medical bills like other unpaid debts. Your hospital --- or any other health care provider --- has three years after billing you to sue for an unpaid debt; after that, it can no longer take you to court. If you make a partial payment in that three-year period, the clock resets, and the hospital has three years from the payment date. If it's a joint debt --- in North Carolina, spouses can be held responsible for each other's medical bills --- your partial payment won't reset the clock for your spouse.
Filing and winning a lawsuit against you doesn't give a hospital, or any other creditor, any money. What it gives them is a judgment that you owe the debt, which your creditor can then enforce by garnishing your wages or placing a lien, or claim on your assets. Under North Carolina law, a judgment must be enforced within 10 years. If you make partial payments on the debt, this won't prolong the time allowed for enforcement.
The statute of limitations affects the hospital's ability to sue you. It doesn't wipe out the debt, which the hospital can keep on the books until it's paid off. The statute also doesn't stop the hospital from turning your account over to a collection agency. Both your hospital and bill collectors are bound by North Carolina law: they can't try to get money by threatening you, cursing you out or publicizing your debt to humiliate you, for instance.
If you file bankruptcy, it will discharge --- wipe out --- medical bills along with other debts that aren't secured by collateral. In Chapter 7 bankruptcy, the court can sell off some of your property to pay your creditors. North Carolina law specifies what assets are safe from sale --- before debts are discharged. In Chapter 13, you'll pay back unsecured creditors over three or five years. If your creditor has already filed a lien on your property, however, bankruptcy can't remove the lien.