Yes they can yes they will - especially if you have made no payments at all (and remember, that interest is still building up). If they take you to court, when they win, the judge will order you to reach an agreement with the credit card company to make payments - and the payments will not be approved if they are token $5 a week payments - think more like $100 a month. If you don't make those payments, the company will go back to the judge who will hold you in contempt and order wage garnishment - which can be up to 25% of you paycheck.
You need to go back and review you current spending habits - get rig of teh smart phone and geta a pre-paid card, ditch pay-for cable channels (HBO, Cinemax, etc.) - basically strip down to the bare bones programming. Check you food spending habits - buying coffee at Starbucks or similar - start drinking WaWa or Shetz coffee and save $2 a cup (that's $10 a week, $500 a year) or make it at home. Stop eating out and learn to cook - buy store brands not name brands. Stay away from the pre-processed food and make you own. Spending right can pay that $4,000 off in 12 - 18 months and teach you how to live within your means.
Grab a second job and use all the pay from that job only for commuting back and forth and the $4,000.