"Possession is 9/10ths of the law" is a homely old aphorism, not a legal principle.
Contra: adverse possession (squatters' rights) do not inhere against government. If somebody camps out in your backyard for 21 years and a day in most states (statutes fiddle with the time periods), the squatter can acquire title by adverse possession.
But if somebody camped out in a national or state forest in 1889, and that plot of land has been continuously lived on by the same family for the succeeding 7 generations, it still belongs to the government, and they don't even have to go to court to evict the squatter. They can send out park rangers or the cops to do it. You never get possession of government property wrongfully taken, no matter how long you sit on it.
Statutes of limitations apply to violations of certain criminal laws, and they limit times for filing lawsuits. So, if these documents were indeed STOLEN from South Carolina's government archives 100 years ago, nobody can be prosecuted for that today. However, legal title does not pass. They still belong to South Carolina, forever. Adverse possession NEVER attaches against the government.
Likewise, the government may voluntarily impose a limit upon itself, such as the 6 year lookback for serious errors on income tax forms. However, there is NO statute of limitations on tax FRAUD. If you intentionally cheated the government out of its money 75 years ago, and the government can prove it, they can make you pay up, with interest and penalties, and put you in jail. (I've always thought that a really clever judicial activist out to "get" the Kennedy's could invent the novel opinion that DEATH does not extinguish the LIABILITY for tax fraud, even though it eliminates the ability to criminally punish the individuals. In that case, the courts could reach back through the probate of the Kennedy Estate and take from Joe Kennedy's heirs the unpaid taxes, plus penalty and interest, for all of that unpaid rumrunning money. Such a ruling would be rightly criticized as judicial activism gone wild. Still, without their money, the Kennedys would be nothing.
Chains of evidence are certainly important once property passes into the possession of the authorities, which was not the case here. Plain old evidence would have to be produced proving that these letters were purloined from the state: proof of ownership.
It seems to me that the proper standard here is that ownership of an heirloom is obvious: presumptively this property belongs to the people who have held it forever. But that presumption can be overturned by demonstrating that, in fact, these documents were part of a state archive and were wrongfully removed therefrom. It would seem to me that the burden of proof would be on the state.