While a recent late payment will damage your credit score, the effect of the late pay will diminish with time, as long as you make it a point to pay the rest of your bills by the due date. Having a recent perfect pay history can begin to overtake the effects of any late payments you may have had in the past. Your payment history makes up 35% of your credit score, so paying on time has a large influence on your credit score.
By co-signing, you agreed to be the backup payer on the account in case the primary folks defaulted (as it appears that they did). If the debt is past due by six years, check your state's Statute of Limitations for debt collection - many states only give creditors 3-4 years to collect on a debt, after which point they cannot bring you to court. A Partial payment will re-set this clock. You may also hit the 7 year limit for how long it can stay on your report (7 years from the date it was first past due with the cable company).
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Making your credit payments on time is one of the biggest contributing factors to your credit scores. Some banks offer payment reminders through their online banking portals that can send you an email or text message reminding you when a payment is due. You could also consider enrolling in automatic payments through your credit card and loan providers to have payments automatically debited from your bank account, but this only makes the minimum payment on your credit cards and does not help instill a sense of money management.