I've been paying my credit card every month at the company's website. They credit me, but never debit my bank account. My bank says the requests are never received from the credit card company (I called a few times). My credit card company says they were paid (I called them a few times, too). What do I do? Just keep paying from my account (the money is in there, no danger of going into overdraft)? Is there a time limit, after which it's too late for them to collect? Or, can they come after me, even 20 years from now?
As far as civil remedies, the statute of limitations on debts would apply; you can check what they are for your state. However, it would be difficult for them to pursue this, as they have already declared the debt paid.
There is also the issue of what form your payment is taking (it's probably ACH, but you should check), and how long your bank will honor it; expiration of debts and expiration of ACH are two different matters.
Things you should do:
Ask your bank whether they will honor these transfers if they are posted later; get the answer in writing.
Notify your credit card company and the bank of the error, and make sure there's a paper trail (e.g. send by email and keep backups of the email).
Meticulously document your displayed credit card and bank account balances; take screenshots, keep paper statements, etc.
(I called a few times).
I would notify both the bank and credit card company in writing that your accounts are not being handled correctly. Ask for an explanation and correction. Set a deadline for them to respond. If registered mail with a delivery receipt is available, use that.
Simply enjoying the fruits of their mistake(s) could be construed as theft or fraud. Try Googling Susan Madakor, whose account was credited with payments intended for the UN!
You say there is no danger of overdraft. Could you cover the total amount not debited correctly in one lump payment?
The person whose bank account is erroneously funding your credit card must eventually notice something. They will recover funds, through their bank, from the credit card company, who will turn to your card account for payment.
For sure, resist the temptation to take some large cash advances on your credit card!