When a paint baby is born, it may or may not SHOW the color gene, but it will carry it (I believe), making it possible for it to produce a colored paint. I don't think people breeding for paints, necessarily hope for a solid paint (not that they would love it any less :) ), but they may end up with one. Even if it is born solid though, it is registered a paint because it has paint lines in either mom, dad, or both. That was as simply as I could put it...:)
Also, if you breed:
Paint w/color X paint w/color, you have 75% chance of paint baby
Homozygous paint X other horse solid or colored, you have 100%
Paint w/color X other breed (not paint) 50% chance of color
Paint w/color X solid paint, 75% color.......
.....I think that is the case, but someone can correct me if I am wrong...It may be a bit more complex. :)