Family Members Believe Nick Carter's Sister Leslie Overdosed?

Sources close to the "House of Carters" reality star tell TMZ ... the 25-year-old sister of Nick Carter had recently moved from Canada to Upstate New York to live with family members while she attempted to kick her addiction.

We're told family members had attempted to rid the home of any prescription medication and were hopeful Leslie was on the road to recovery.

Sources say family members believed Leslie would be OK to stay alone in the home for roughly an hour on Tuesday while they ran errands ... but when they returned, Leslie was unconscious.

Officials were called to the scene ... but Leslie could not be saved.