Statistics vary, but most show that it's really on a very small percentage (around 1 to 2 percent) of people with chronic pain who do become addicted to pain killers. However, people can - and do - develop a dependence on certain drugs.
According to the FDA website, someone who is addicted is someone craves and uses the drugs compulsively, has to have them, has to use them. Someone who has become physically dependent on a drug is someone whose body has adapted to the drug and it's not as effective any longer. People who are dependent on opioids do go through withdrawal symptoms when they stop taking the drug, but this doesn't mean that they were addicted. It's just that their body needs to get the medication out of their system. In this case, the symptoms are physical, rather than psychological, as with addicted people.