Turns out the backstory is a lot more involved than that. Apparently someone the songwriter knew had some manner of mental illness - agoraphobia, social anxiety, something in that vein. (It's been years since I've heard the story, and I'm going from memory, so sorry if the details are a bit off). This person almost always stayed inside and avoided most human contact, but they still got sad whenever there wasn't rain - you see, when there was rain, they had an excuse to stay inside, and nobody would fault them for it. When the sun was shining and it was nice out, they'd still be inside, but now they'd be standing out and drawing attention to themselves. Something that was a cause of happiness to most actually made their troubles even worse, because it was something they couldn't share.
I've been pretty lucky in my life, all told - I've had problems, of course, but I've gotten through them, and there's been nothing so bad as that. Others aren't so lucky. The weather we've had recently is what got me thinking about this song because good lord was that a lot of rain last week, but the song itself is about a lot more than the weather. A lot of people have those hidden depths to them as well, and folks who go blundering about on the surface(a group I'm occasionally a part of myself) can cause a lot of pain without even realizing it. I've got a mostly-written post in the hopper on a related topic, and a couple more I'm thinking about, but I figure I can at least hint at it through song. And it is still a nice song, even if it's not a happy one.