To Sanders and Warren, New Hampshire is a must-win, given that they represent neighboring Vermont and Massachusetts in the U.S. Senate. But the student-debt issue is a microcosm of the challenge both candidates face selling their progressive vision in places where more moderate views prevail.

A recent YouGov/CBS poll showed that 61% of Democratic voters in New Hampshire want tuition lowered through added government subsidies, but not free. Only 32% of New Hampshire voters favored tuition-free colleges, while 7% said higher education should cost students whatever the market allows.

"When you put together free and college in the same sentence, that's where you might see some disagreement among voters in New Hampshire," said Dante Scala, a politics professor at the University of New Hampshire. "It's fool's gold to chase younger voters if at the same time you're turning off a greater number of older, more moderate voters."