At the point of saturation, no more moisture can be added to the air and so no more heat can be absorbed by the process, so the air fanned out the unit now stops feeling cold.
When the starting humidity is already high, as in areas near large bodies of water (lakes, large rivers or the ocean), there is already little room for adding any more moisture into the air, so an evaporative cooler doesn't have the latitude to produce its most effective output. In other words, it just can't create enough air coldness.
People living in these areas that buy swamp coolers are already doomed to disappointment because the units simply cannot create an air chilling effect. And so buyers send the units back to the manufacturers complaining their product doesn't work!
On the flip side, if you live in an area with low humidity, there is a huge volume of dry air just waiting to absorb the moisture from one of these devices. The only way they are going to see the effectiveness reduce is by running a swamp cooler in an enclosed space with no ventilation (to the outside).
What happens then is the same as above. As the unit pumps more and more damp air into the atmosphere, the room's humidity rises until it reaches saturation point.
Of course, this is easily remedied. Simply by opening a window and a door a crack and allowing a cross-flow of outside air to circulate through the room will help maintain the dry air in the room and the cooler will keep working just fine.
The unfortunate fact of this story is that those living in dry climates can take full advantage of the low energy consumption and eco-friendliness of evaporative cooling devices, while those living in humid areas cannot. It's like a zip code lottery whether you get to save money or spend more on rising electricity costs because you need air conditioning to keep you cool.
And there is no other alternative that is quite as effective or convenient.
If you want to read more about how evaporative coolers work and where they are best deployed, visit this website (Ventless Portable Air Conditioner):