Not all debt is created equal -- and not all debt is bad. It's perfectly reasonable to carry a mortgage, a car loan, etc. (After all, when was the last time you bought a house with cash?) You simply need to pay attention to the cost of the debt. If you're carrying revolving debt on a credit card that's charging you 18% per year, you're in a bad situation. If your student loan is costing you 7% per year, that's much less worrisome.
Another consideration is what else you might do with the money you'd use to pay off a low-interest loan. Imagine that you've borrowed $5,000 at 6% and you now have the money to pay it off in full. You could do so, but consider the alternative. If you're bullish about a stock or two -- and are fairly sure that, over the next five years or so, you'll earn at least 15% on them per year, on average -- then you might choose to keep the loan and pay it off gradually, as you originally planned. You might take the $5,000 and invest it. If the stocks perform as expected, you'll be earning more than you're paying out in interest. Even if they only return 10% per year, on average, you'll come out ahead -- because you intelligently built in a margin of safety.
That's why mortgages, for example, are not necessarily a bad thing. If your mortgage rate is low, it makes perfect sense to keep paying it off gradually. (If your rate is high, consider refinancing it, if you can!) Mortgage interest also brings some tax benefits with it.
Along the same lines, investors shouldn't assume that any debt on a company's balance sheet is a bad thing. If the company has borrowed funds at 7% and has put that money to work earning 12%, that's a positive. Too much debt can be trouble, though. (This Whitney Tilson article urges us to not forget debt.)