No, unless you have a history of substance abuse, having to increase the dosage of your pain medication doesn't mean you're becoming addicted. Doctors are usually careful to start people
on pain medicines at low doses, with a plan to slowly increase the dosage as needed until, together, the doctor and patient find the right dose needed to control the pain. Some people need only a low dose of pain medicine, but others need higher doses; that's just the way some bodies and some types of pain are. Since it's usually not safe to start with a high dose right off the bat, a period of slowly adjusting the doses upward should be expected.
For some people and some medicines, there's also an element of building up a tolerance, which means the body gets used to the medicine, so that higher doses are needed to get a certain painkilling effect. This is not the same thing as becoming addicted. Addiction means that a person has cravings for the drug and engages in behaviors that are counterproductive to happiness and success in order to get it.
Finally, many people have to increase the dosage of their pain medication because the condition causing their pain is getting worse. Common examples include worsening arthritis or the progression of cancer.
Many people get confused because when they don't take their medication, the pain comes back. Then they think they're addicted to the medication simply because they need it. But this isn't addiction either. Yes, they're dependent on the medication; they need it to control their chronic pain. All that this means is that they're in pain without it. Their need for the medication won't go away because their pain doesn't go away.
That being said, it's an unfortunate fact that a small minority of people taking painkillers will become addicted. These are usually people who have had problems with substance abuse before starting pain medicines. Signs of addiction include cravings, getting pain medicines from several different providers, and being dishonest about the medication with family and physicians. If these signs come up for you or your loved one, get professional help as soon as possible.