Since all payments are processed by a third part secure payment gateway, we don't have many details about payments that fail. Usually we receive a standard message saying "Credit card declined by issuer. Contact your bank to determine the reason".
Based on our experience though, most of the times it's related to anti-fraud filters based on location. We can list some situations that certainly will cover 99% of the situations:
Billing address in the credit card company doesn't match to billing address in our system.
Solution: update your billing address.
Credit card used outside of the country where it was issued (for non-USA residents).
Solution: contact your credit card company to allow making purchases beyond your boundaries.
Solution: make a payment to your credit card company and clear balance.