This could be possible depending on the size of the island and the type of trees in the forest.
Forests have been shown to play an important role in the formation of clouds and in the stability of rainfall. Trees suck up water from the ground in large quantities and most of that water is exhaled right out again through the leaves. Vegetation stand for as much as 90% of all moisture in the athmosphere and trees produce much more than smaller plants.
Forests also release tiny airborn particles like dust, bacteria, pollen and fungal spores into the air that promote condensation and gather the resulting moisture, hastening cloud formation
Research on deforestation shows that rainfall decreases and cloud formation lessens when trees are cut down. In similar ways rainfall can be increased through the planting of new forests.
But you're on an island, so what stopps the moisture from just drifting off due to winds?
If you could considder having mountains in a half moon shape on your island it could both help keep the local moisture put, but also contribute more moisture due to condensating moist air blown in from the sea. This would require the opening of the half moon shape to be facing the most common wind direction and that the most common wind direction blows in from a warmer area.
Then with the right conditions I think you could very well have about 24/rain with some seasonal variations. For instance, if the winds turn for a long period the cloud cover might thin out. Or if the wheather is colder the winds might bring less moisture.