Yes. Most credit card companies have a hardship program. Banks will temporarily lower your interest rate for about 6-12 months to help you get back on track. Sometimes, they may lower your payment as well. Your financial hardship must be compelling to merit the benefits.

Contact your creditors and tell them about your personal hardship. Your story must be persuasive. Provide specific details of your income and expenses. The hardship reason should involve loss of income and/or increased expenses in essential categories. Your budget should not include categories such as travel, entertainment and eating out. These should all be $0 per month to qualify.

Contact Amex right away to explain the reason why you can't pay. See if you qualify for their hardship program. Consider credit counseling to help with Amex and other creditors.

Late Fees: Once your payment is one day late, you will likely be charged a late fee.

Credit Effect: When you reach 30 days late, this will negatively affect your credit score.

Closed Card: Your card will likely get closed once you become delinquent.

Charged off: Typically, once you fail to make the minimum payment for 6 months, your account is charged off

Lawsuits/Wage Garnishment: Your creditor may sue you for failing to pay your debt. The may attempt to garnish wage as well.

There is no documented information about a Debt Relief Program thru American Express or whether or not they will negotiate debt.  However, they do offer an internal hardship program. And American Express also works with credit counseling agencies to lower your interest rate and payments.