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No, you're not financially responsible for the debt because it was incurred when you were a minor.
You can voluntarily agree to pay it, but they can't sue you for it because of your age at the time the debt was incurred.
If the treatment came from a state, city, county or other government owned hospital, there is no statute of limitations, so the debt can hang around forever.
You can write them a letter denying that you're responsible for the debt and also demand that they cease contacting you about this uncollectible debt under the Fair Debt Collection Practices Act.