A pair of small Pennsylvania insurers focused on long-term care could soon become one of the nation’s costliest insurance failures ever, highlighting the widespread problems that have plagued the industry niche for more than a decade.

Two insurance units of Penn Treaty American Corp., which have combined assets of about $600 million and projected long-term-care claims liabilities topping $4 billion, are on track to be liquidated early next year, according to filings in a state court in Harrisburg.

While...