I'm really bad about not wanting to set my tarp up. I often push my luck and hope any rain will hold off until morning or not happen at all. Most of the time I get away with it. However, I have had showers come through in the night. Sometimes I just wait it out if it doesn't look like its really going to do more then just drop a few drops or lightly rain for a few minutes since the biy will protect me. If it looks a little longer lived or its getting near the time to get up, I just grab my tarp and throw it over me like a blanket until it stops or I get up. A few times, I moved under a nearby tree with a thick canopy (or just choose to camp under one in the first place) and went back to sleep, the rain didn't pentrate much and the bivy though slightly damp, kept me dry. If the rain started near dawn, I figured setting my tarp up that that point was a waste of effort and just packed up early and hiked a little in the rain with my flashlight until dawn. Otherwise, I actually have to get up and set my tarp up in the dark. Fortunately I haven't had to do this more then once or twice. Though if I'm sure its going to rain over night or its already raining when I get to camp, obviously I just set the tarp up.
Tarping in the rain: Pick a camp spot where water won't pool. Any site that has been heavily used will pool water since years of campers will have compacted the ground there into a shallow bowl shape and the harden ground won't absorb water very fast. Pick terrain where you can see how the water will flow away from you. I like slightly sloped ground. Try to find sheltered sites from wind like behind boulders or in the trees or shrubbery. If there isn't a lot of wind, you can set your tarp up high and wide to maximize space under it. The stronger the wind is, the lower down you have to stake the tarp; or at least the lower you have to stake the sides of the tarp which makes it narrower and shorter. If you know the wind is only going to come from one direction, you can put that side down low while leaving the opposite side up higher to give more room.
Bivy in rain under tarp. A water resistant bivy is not water proof. It will resist water for awhile but eventually it will saturate and let the water soak through to your sleeping bag. So if you are only getting a little water on it or very light rain, there isn't a problem. It won't handle prolong rain. Most of the time, the bivy sack won't get wet if the tarp is properly setup. But there will be times where it does get wet when using a smaller tarp like my solo size one due to wind. I once had a small amount of water pooling at the foot of my tarp but the bivy kept my sleeping quilt dry since it wasn't too deep. I've had mist and light rain blow under my tarp due to the wind shifting and it got the bivy lightly wet. At that point you have a few choices. Can you ignore it and remain dry (ie. its not happening enough to worry about)? Do I need to get up and change my tarp setup (lower it or rotate it)? Or can I block the rain with something. Like using my pack with packcover, or using that reflective umbrella I brought for desert hiking, or using my rain jacket hanging off my pole somehow. I sometimes set the head of my tarp tied to a large tree instead of using a treking pole for that very reason since the tree will block the wind and rain from that direction. I normally stay dry or only slightly damp in rainstorms.
However, I did have 1 time recently (2012) where I got very damp. The water didn't soak all the way through my sleeping quilt but the top part got very wet. Read here: http://www.postholer.com/journal/viewJournal.php?sid=dbda992b7f98075a5cb31864de29e90a&entry_id=34815
What I learned from it was to not camp on tent platforms that are raised up in the air more then a few inches. Well wooded tent platforms suck when using a non free standing shelter anyway, but I didn't want to deal with all the wet leaves on the ground at the time. Big mistake. So even after years of successful use, you can still find that there is more to learn.