Yes, psychologists tell us it certainly is possible to become addicted to almost any activity -- eating, shopping, the Internet, even exercise. When that happens, something that may be harmless or even good in itself becomes harmful, because it takes over our lives and controls us.
Addictions of any kind can be hard to break; one reason is because we must first admit we have a problem. But that's often hard to do, because the habit has become so much a part of our lives that we can't imagine living without it. Pride gets in the way also, making us refuse to admit that what we're doing is wrong. We fail to see that we're not only hurting ourselves but also those who love us and need us.
What can you do? First, pray for your cousin, that God will help her realize she has lost control of her life at this point. Pray too that God will help her understand that she may be using this to escape from deeper problems in her life. Then enlist family members who can encourage her to face this problem and get professional help if she needs it. The Bible says, "Carry each other's burdens, and in this way you will fulfill the law of Christ" (Galatians 6:2).
Most of all, encourage your cousin to open her heart to Jesus Christ and put Him first in her life. Down inside each of us is an empty place that only God can fill -- and He will, as we turn to Christ in repentance and faith.