Yes, and no. 'I can't deal with it' means the same as "I can't cope with it.' But: 'I deal with many situations of that nature.' means more than just coping. It means you manage/handle all the details and aspects of those situations.

I agree with Karen. I would add that “cope with” often has a psychological/emotional aspect — i.e., having to do with whether the person handle a situation from a psychological/emotional perspective? In contrast, “deal with” often has more of a business-like aspect — i.e., having to do with whether the person has the knowledge or skills to solve a problem.

I agree with John. Maybe a couple of examples will help: I can't cope with my children. They are driving me crazy. They are stressing me out. I think I'll have to send them to their grandparents for a few weeks. I can't deal with my children. Their behaviour is out of control. They don't listen to my requests. I don't know how to get them to do their school work. etc. Maybe I need to buy a book on parenting skills.