The family that owns a drug company is now being sued over the toll of opioid painkillers in one New York county — and it's likely to be sued by hundreds more.

Well over 1,000 lawsuits filed by state and local governments blame drug companies for a crisis of addiction and overdoses across the country. Only a few of them have named members of the Sackler family, which owns and controls Purdue Pharma, as defendants.

But a new filing by New York's Suffolk County asserts that family members participated in billing OxyContin as non-addictive even though they knew that was not the case. The claim la...