Paul Mawhinney, a former music-store owner in Pittsburgh, spent more than 40 years amassing a collection of some three million LPs and 45s, many of them bargain-bin rejects that had been thoroughly forgotten. The world’s indifference, he believed, made even the most neglected records precious: music that hadn’t been transferred to digital files would vanish forever unless someone bought his collection and preserved it. Mawhinney spent about two decades trying to find someone who agreed. He struck a deal for $28.5 million in the late 1990s with the Internet retailer CDNow, he says, but the sale of his collection fell...