SCRANTON, Pa.—Four decades ago, Jimmy Pliska says, he was sexually assaulted by his local parish priest on an overnight fishing trip. Now, he has an agonizing decision to make.

Amid a recent wave of sexual-abuse investigations and allegations against the Catholic Church, Mr. Pliska wants to sue the Diocese of Scranton, which employed the priest. But the case is too old to bring to court. Although state lawmakers have proposed lifting the statute of limitations on the sexual abuse of children, it is unclear when—or if—that...