While not all CC companies are the same, here is a high level overview of what happens.
Thirty days past due(one payment late), a friendly reminder letter, but probably no call-unless they have previously deemed you high risk by previous payment history or FICO score.
Sixty days past due, more letters, and they will throw you in a dialer and make calls.
Ninety days past due, more letters, account cancelation, and an increased level of collection calls-your really on their radar now..
120 days through 179 days past due...more letters, more calls, and your account has probably been assigned to a specific collector to work. That is when you really start running into insensitive, uncaring representatives of the bank.
180 days past due-charge off-Credit card company officially posts the account as a loss.
From here it really varies bank to bank, but in summary, your account is either turned over to an outside collection agency(sold), or to "in house" legal. They will continue to try to collect for a while. If they can't collect,then they will serve you, and get a judgement against you.
If you think you will be able to avoid bankruptcy, then see if they will work with you. If bankruptcy is on the horizon, then ignore them.
Just an opinion...