Increasing a dosage of a medication doesn’t necessarily mean that a person has become addicted to it. The human body has an amazing ability to adapt and sometimes, adaptation means getting used to or tolerating a certain dosage of medication. What can happen is that the pain level of someone taking an opioid, or narcotic, may be relieved at one point, but as the body becomes used to the medication, it no longer does the job and higher doses are needed.
In your mother’s case, it is possible that the medication she is taking is not really the right one for her. If she is still having pain, it may be a good idea for her to speak with her doctor about other pain relieving options.