They do in the UK. If you don't acknowledge a debt for 6 years, and during that 6 years the creditor doesn't take legal action against you, the debt becomes unenforceable - statute barred debt is the technical term.

The debt still exists, they can still harass you about it, but you cannot be taken to court and made to pay.

Now, in most cases, that doesn't happen that frequently, as most creditors would take you to court well before the 6 years are up.

Details vary for jursdiction to jursidiction and between different types of loans; http://www.consumerfinance.gov/askcfpb/1389/what-statute-lim...