Just because a account is written off, doesn't mean they can no longer attempt to collect the debt. They can continue to attempt to collect the debt, even taking legal action to do so until the statute of limitation has expired. The Statute of limitation is determined by the date of last account activity, usually when the last payment was made. The SOL varies by state, so you would need to check your state law to find out what the SOL in your state is.
Even after the SOL expires they can
TRY to collect, however they cannot sue or take any legal action to collect it. Basically all they can do is call and write and try to intimidate you and hope you'll cave and send them money. Often collectors who are collecting on Time Barred debt buy the debt is portfolios for two to five cents on the dollar, so if they buy a hundred dollar debt for a buck, and can get you to pay face value, then they just realized a 90% profit.
After the SOL has expired, you need to be very careful what you say or do so as to be certain you don't inadvertently restart the statute of limitation.
After thought:
The SOL on reporting the debt is covered by the FCRA, and that SOL is not changed by virtue of selling the debt (that would be considered re-aging the debt which is illegal). 180 days from the initial default date is used to determine when a debt can continue to be reported or not.
It is also important to remember that Just because a Agency (for example Asset Acceptance) buys a old debt, they are STILL considered a "Collector" under the fdcpa. This is clearly stated in FTC staff opinions to Arbuckle/Midland Credit Management available on the FTC website. Because they purchase a debt does not mean they are considered an original creditor.