Do unpaid medical bills sent to collections ever go away without payment?

The non payment of any debt can result in a civil suit by the creditor. However, all U.S. states have SOL's which determine when a creditor can no longer pursue the debt through legal procedures.

Please be advised, when the SOL applies it is the debtor's responsibility to bring forth that reason as a defense. Likewise, a creditor/collector can still pursue collection even after the expiration of an SOL by the usual methods, phone calls, letters and so forth. Even so, the debtor has the recourse to send the creditor/collector a letter of "cease and desist" which prevents future debt collection contact. If such a letter is rendered the creditor/collector must cease all contact with the debtor unless/until a civil suit is filed.