How could air that is saturated with water become oversaturated without adding any water?

By cooling down the air

Explanation:

Air saturated with water can become over-saturated without adding water when it is cooled down. When the temperature of the air is reduced, the water holding capacity of air is also reduced.

The air at higher temperatures at hold more water as compared to the one at the lower temperature. This is due to the fact that water vapors are present in warmer air. Cooling down the saturated air leads to condensation of water vapor and makes it over-saturated.