Can the IRS make you pay for back taxes than are more than six years old?

Absolutely. The Statute of Limitations is on many things...first..on audit and then on assessment. The tax must be assessed within a certain period, and if the amount found is more than 25% of the tax you paid, there is no limit on time (as that's considered fraud). No SOL even starts to run until a return has been filed. Don't file and your perpetually open to assessment and collection. Also many things "toll" the counting of the time, including receiving a notice or request. Ignoring it does not (and should not) work to your advantage.