While a major public-health campaign has had some success in reducing the number of people who take potentially addictive narcotic painkillers, those patients who are prescribed the drugs are getting more of them for a longer time, according to a new study.

Nearly half the people who took the painkillers for more than 30 days in the study’s first year were still using them three years later, a sign of potential abuse.

The report, released last week by the pharmacy benefits manager Express Scripts, found that nearly 60 percent of patients taking the painkillers to treat long-term conditions were also being prescribed muscle relaxants or anti-anxiety drugs that could cause dangerous reactions.

The study looked at the pharmacy claims of 6.8 million Americans who filled at least one prescription for an opioid between 2009 and 2013. Opioids include commonly used drugs like codeine, morphine, oxycodone and hydrocodone.

“Not only are more people using these medications chronically, they are using them at higher doses than we would necessarily expect,” said Dr. Glen Stettin, a senior vice president at Express Scripts. “And they are using them in combinations for which there isn’t a lot of clinical justification.”

Overdoses involving prescription drugs are a leading cause of accidental death in the United States, and opioid painkillers play a role in about 70 percent of such cases, according to the federal Centers for Disease Control and Prevention. Opioid overdoses led to 16,000 deaths in 2012, the agency said.

The study found that nearly one in three patients were prescribed an opioid and a benzodiazepine in the same month, and around the same percentage were prescribed a muscle relaxant and an opioid at the same time. About 8 percent of patients were taking all three types of drugs – a combination known as a “Houston cocktail” which gives a heroinlike high – during the same period. And 27 percent were taking more than one opioid at a time, another hazardous combination.