When you stop paying your credit card bill the bank will want to find a way to get you to start paying again. The first step will likely be making phone calls and sending letters. If that does not work the bank may hire a debt collector to turn up the heat on you.
The collection agency will call, a lot. And send many letters. If still you are not paying your bank will likely hire a local debt collection law firm to attempt to collect.
The law firm will likely again send more letters and make more phone calls. If still not payment is made then they are likely to bring a lawsuit against you.
In order to start a law suit the bank will have to file a complaint and serve a summons on you.
In my experience it typically takes 10 - 12 months from the time of the first missed payment and the time of the law suite.
Filing for bankruptcy will likely stop all collection calls, letters, and law suits the moment the case is filed.