Smaller birds are more prone to this than larger ones, and only if they stay in the rain too long.
In heavy rain, or if a bird can’t find any shelter, the water will eventually get in to the air pockets under their wings. These then get full of water and will cause the body temperature of the bird to fall. This can be enough in itself to cause hypothermia, but if the bird is also unable to dry out properly, the prolonged cold water next to their body will eventually cool them too far and hypothermia will set in.