former painter here.

it is true that the moisture itself is not a factor unless it can somehow get into the finish...but it happens sometimes because some low-budget painters paint in tents or in a part of the building that is exposed to the outside - the reason for this is that the best way to check a color match while you're painting is to look at it in direct sunlight.

another factor about rainy days is the temperatures. it is much much easier to do a perfect job on a hot and dry (low humidity) day - when it's colder the paint doesnt flash as good, so it tends to run. you also have to use low-temperature solvents, which do weird things if you're not used to working with them.

third, when the job is drying you want it to "flash", or have the outer layer dry and harden, as quickly as possible, so that the lint and dust that settle on the car while it's parked have no time to get stuck in the finish.

finally, rainy days are dark and cloudy, which make you have to rely on artificial light, which as i mentioned is not as good as sunlight for a painting environment.

in california the temperatures during the rainy season are 15+ degrees F lower and i used to hate coming to work because i had to redo my work more often because of those issues.

so the answer is, you can get as good a job painting during a rainy day as a sunny day, but it's harder to do so because the margin of error is higher.