The word "saturated" on its own can be confusing. Without getting too technical on the gas laws and vapour pressures, it is will normally suffice to look at it as follows :
At 100% relative humidity the air is saturated as regards evaporation, that is to say no more water vapour will be absorbed into the parcel of air. You will have reached the wet bulb temperature.
You would still have to cool it slightly more to the dew point, for it to condense out as visible moisture. ie cloud, fog etc. That is why the dew point is colder than the wet bulb temperature under most circumstances.