I recall a Japanese econ grad student telling me that in fact real interest rates were negative for some span and people were STILL saving in the late nineties in Japan. He also blamed several bubbles at the time (notably, real estate in Japan) on this. Interesting if true... (Anyone know the details and can they justify the connection without resorting to framing biases of investors?)
Fred, I think you answered your own question -- even if many people would save some of their money even if the risk-free interest rate were zero, the quantity of funds demanded by borrowers at this rate would exceed the amount saved. So, a positive interest rate induces some people to defer consumption (save) and others to borrow less, until the market clears.
And this would be the case even if there were no risk of nonpayment, and no inflation/deflation.
By the way, there have been times and places where the measured real interest rate was essentially zero; I think this happened in Japan in the 1990s.