Painting during rain is only possible with interior painting. During humid weather interior paint will require more time to fully dry and cure.
For exterior painting you will have to wait until the weather clears and the surfaces(s) are dry. Usually, 4-5 hours of drying with good temperatures is enough for exterior paint before it rains.
Oil paint is especially risky if in doubt that the surface is dry.
If you're doing an exterior and using an oil-based paint, it would be wise to wait until you are certain the area is dry. It's always important that the surface is dry, but latex paints will breathe a bit and give you a little leeway if you've painted too soon. But still not recommended at all. Oil-based paints will most likely peel right off if it's damp.
On an interior, I've been frustrated by slow drying paint when it was extremely humid. Fans seem to help.
A few things to keep in mind: Having wet paint on a surface for an extended period of time can sometimes pull stains from inside the substrate that may not leak through if it had dried quickly. Quick dry primers are available, but are not as good if you're trying to cover nasty stains. Also, slow drying exterior paint (more often oil-based) leaves more time for mildew spores to attach itself, which can kick start mildew growth on a house.
There are moisture meters available, though I have never used them. And I think the acceptable level of moisture is 10-15%, but don't quote me on that.
Temperature can also be a problem in rainy season. If Temp is to low it can cause sagging due to slow drying process.