While this used to be a theoretical concern (and most people felt that it would never happen), interest rates in Europe have started to dip below zero. The two month Euribor is currently at ! The Wall Street Journal wrote an article explaining some effects of this.
Here is some food for thought:
1) What does a negative interest rate mean?
2) If you borrowed money from someone, do you get to repay them less in a year?
3) If you have a bank mortgage, does this mean that they should pay you for the privilege of loaning you money?
4) If you had money saved in the bank, should that amount decrease over time?
5) Should you go out and spend all of your money today, since it would be worth less tomorrow?