However, it is hard to imagine that interest rates would increase without a serious uptick in economic performance. Such an uptick would automatically go hand in hand with increased tax revenues and lower spending (because social safety net spending would shrink automatically).
Basically, any scenario in which interest rates grow are scenarios in which automatic stabilizers will reduce the government deficit in other places. It's a healthy system in which the feedback mechanism go in the right (i.e. stabilizing) direction.
"However, it is hard to imagine that interest rates would increase without a serious uptick in economic performance."
It's not that hard to imagine; what you describe is essentially Stagflation. You may consider it unlikely, but there are some known potential causes[1].
As I understand it, Stagflation happens when increased demand is less able to stimulate increased supply than one might expect.
As mentioned in [1], an oil supply shock could be a cause. That is not outlandish given the current instability in the middle east and our tense relationship with other oil producers (Russia and Venezuela). The US and Canada do produce a lot of oil, which may offer insulation, but I don't think that's necessarily a defense.
Another cause listed is tough regulatory atmosphere. For instance, if the EPA decides to strongly curb CO2 emissions, or misguided labor laws come into effect, or the healthcare system in the US gets even worse. Again, not outlandish.
Ah, right. I still think there are two mitigating factors here to what I said.
First: In the stagflation of the 1970s (which is really the only significant empirical data point we can draw from), increased interest rates were a political choice made by the central bank rather than an economic necessity.
Second, and more importantly: Stagflation is characterized by inflation, which means that nominal GDP grows even while real GDP is stagnating.
Since real GDP is irrelevant to the debt-to-GDP ratio (witness the debt-to-GDP ratio through the 1970s), the conclusions for whether one should worry about the debt-to-GDP ratio remains the same as far as I can tell.
Why is that hard to imagine? If, hypothetically, our economy fell off a Great Depression style cliff, wouldn't that make it harder to pay our debt, expectation of which would force rates up, and be somewhat self-fulfilling?
Basically, any scenario in which interest rates grow are scenarios in which automatic stabilizers will reduce the government deficit in other places. It's a healthy system in which the feedback mechanism go in the right (i.e. stabilizing) direction.