I was there around the same time as 3TinksAndAnEeyore and I agree with that assessment--Saturday during the DAY appeared to be as or more crowded than Wednesday and Thursday, and despite my best efforts, I got wet feet too.
It wasn't bad and DCA in particular was eerily empty in the evening, but unless you adore Toy Story Midway Mania more than anything and want to ride it again and again, I don't know that it was really a significantly better day than Wednesday or Thursday of the same week. After all,
Disneyland and DCA are made to absorb a moderate number of people with minimal lines. (TSMM was basically a walk-on that night--probably half the guests in the park were in the queue or riding! Soarin' wasn't even filled up all the way when we rode; I think a full third of the theater was empty. A Bug's Land was deserted; we spent 5 minutes there and there were a few people in line for Heimlich's train and a cast member standing all lonely by the bumper cars, and then as we were leaving, a small group (no more than 4 people) came in. That was it. It would have been just fine with a few more people.

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If you just want a dead day, then a weekday in the "off season" is great. For rain, you can check a weather site like
http://www.weather.com/outlook/recreation/outdoors/wxclimatology/monthly/graph/USCA0027?from=search --on average, February appears to get the most rain, and it also has the fewest days.
Even if you CAN make your decision of when to go just a few days in advance, you still can't necessarily be sure. When we went, the forecasts for even a day before generally predicted rain for Thursday through Sunday. In reality, it only rained Friday afternoon/evening and most of the day and some of the night on Saturday. We did all our sunny day stuff on Wednesday and then Thursday since we thought Thursday looked clear and one of the three sites we checked said it would be clear (and it was).
So anyway, no, there's no way to tell for sure. I don't blame you for wanting to see a rainy day for yourself but with weather, there are no guarantees. (Though when he was younger, my brother wanted to be a weatherman because he thought they controlled the weather. Sssh.

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