The relationship between sawtooth events and magnetospheric substorms has been discussed extensively. However, the relationship between sawtooth events and magnetic storms has not been systematically examined. Using the sawtooth event list and magnetic storm list from January 1998 to December 2007, we investigate whether sawtooth events are storm time phenomena and whether there is a dependence on the strength and phase of storms. We have found that most of sawtooth events occur during storm time. Nevertheless, there are still 6 sawtooth events (5.4% of total events) that occur during nonstorm intervals. Sawtooth events also tend to occur during intense storms, with an occurrence rate of 63.5%. Sawtooth events can initiate during any stage of storms, however 55.9% of sawtooth events occur during the storm main phase through the time the ring current reaches its maximum strength. Therefore we conclude that sawtooth events are very often but not necessarily storm time phenomena. And not all storms contain sawtooth events. We suggest most sawtooth events occur during a special subset of storms that have just the right driving conditions to set intense, periodic, near-tail magnetic reconnection bursts.