Yes, it is possible; some people would rather have our sympathy than our help, because they actually enjoy the attention it brings them -- and they know they might not get it otherwise.
Is this the case with your cousin? Perhaps it is -- although she may not even realize it. But there may be other explanations for her refusal to face her problems and do anything about them. For example, many people wish they could get rid of their problems or their bad habits -- but they never do anything about it, because they don't want to go through the hard work of conquering them. The Bible warns that "fools despise wisdom and discipline" (Proverbs 1:7).
Let me suggest that the next time your cousin shares a problem with you, ask her a simple question: "What do you think you ought to do about this?" Don't let her get by with saying that she doesn't know; help her to think through an answer. Not only will this encourage her to realize her need to take practical steps, but it may also cause her to face her own lack of discipline or willpower.
Most of all, encourage your cousin to turn her problems -- and her whole life -- over to Jesus Christ. God knows her problems, and He also knows the solution to them. More than that, He can give her new strength and a new determination to overcome them and become the person He wants her to be.