"But, in theory I just don't get it. Say we get a week of almost non stop rain, with no sunshine. If you were to artificially create that environment, It would not be good for your tree in it's pot."
The degree of artificiality is on a sliding scale. Bonsai are in the local environment, same as landscape trees, receiving similar amounts of light, humidity and air circulation as those trees. They're not on Mars.
"To create the environment i'm talking about I would just continue to water my tree non stop for a week under complete shade. I know this would not be good for the tree, so i'm wondering if it would be better to put my trees out of the rain."
You assumptions are a bit exaggerated. Just because its raining, things don't die. Hasn't worked that way for several millenia. Even if the soil is soaked by a week of rain, it doesn't mean the roots are drowning, especially if the soil drains. Additionally, you assume the tree is not getting ANY light under clouds. Both assumptions are completely false. Light is a relative thing, as is "too much" water. Sure, the extremes of either will kill trees, but the distance between "excessive" and "extreme" can be a long one.
Your soil drains water well from the sound of it. The water draining through it does not leave a vacuum behind. Air is drawn into the soil during rainstorms. If your soil drains extremely well, then you soil also has alot of air being drawn in as well. Even if the soil doesn't drain exceedingly well, if it's up off the ground and its drainage holes are clear, the roots aren't going to drown and rot in a few days.
Also your assumption that root rot can set in within a few days is erroneous. Root rot isn't due to too much water. It can be brought on by a number of things. Repeated overwatering of inadequate soil sets the stage for root rot.
Your trees aren't growing in complete shade when it clouds up and rains. They are getting light, more light than they would get inside even under flourescent lighting set ups. Even prolonged cloudy periods can't kill trees--the effects are variable species to species, but rest assured, a cloudy rainy spell isn't going to kill your trees.
If rain on your trees upsets you, simply angle the pots, so most of the water runs to one side (Which can, ironically set up a situation where your soil won't drain-if you've got a pot with limited drain holes which aren't located on the downhill side of the slant) or get them under the eaves of the house. It's mostly unnecessary, though.
Again, I refer you to the trees in woods near you...