In Utah, yes:
Utah Code § 78B-15-109. Limitation on recovery from the obligor.
The obligor’s liabilities for past support are limited to the period of four years preceding the commencement of an action.
As to the period of time in which one has to enforce an existing child support award:
Utah Code § 78B-5-202. . . . Child support orders.
(6)(a) A child support order or a sum certain judgment for past due support may be enforced:
(i) within four years after the date the youngest child reaches majority; or
(ii) eight years from the date of entry of the sum certain judgment entered by a tribunal.
(b) The longer period of duration shall apply in every order.
(c) A sum certain judgment may be renewed to extend the duration.
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