Nope. There is no resetting the clock. The 7-1/2 years is from the date the account first went delinquent. A subsequent payment doesn't reset the clock.
That said, you could make a payment that brings the account completely current. Then a few months later you go delinquent again. That new delinquency will have its own 7-1/2 year reporting period. But partial payments that don't pay off the entire past-due amount have no effect on the reporting period.
Paying an out of date bill does indeed reset the statute of limitations in many states and does allow it to update your credit report with more lates up to the current payment.