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Here's the problem. I was wearing my bi-focals, and they create an optical illusion that makes straight lines look crooked. I realized this after tapping the first two of the holes. They were at a slight angle. The last two of them went straight in. The end result is that the plate fits perfectly, as long as I have only 3 studs installed. I install all 4 and they don't locate properly.
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How much off-straight are the threads? Also, does the chuck mount to the backplate with studs (threaded both ends) or capscrews? If the holes aren't too much off-straight, you might want to try using a pulley tap (tap with extra-long shank ground to the clearance diameter of the appropriate screw). Mount the chuck to the backplate using the "good" holes, with the "bad" onces roughly halfway between one lining up and the other lining up, and tighten the 2 screws really well. Then, re-tap the bad holes using the pulley tap (it'll guide into the start of the threads) - the "brute force" of the tap being held in line by the clearance hole for the screw is frequently enough to be able to get screws in. We do this fairly often at work.
If it's still a bit too much, the next step is "Michigan screws" - you've gotta make your own, to fit the application. They're capscrews that are ground to a smaller diameter starting just below the surface of the threaded plate, and ending about 1/2" below the head. The reduced diameter lets them move a bit more to the side than a full-diameter screw.
It's too late now, but you should get yourself a piloted tap wrench (KBC part numbers 1-397-101, 102, and 103). After drilling the holes, you leave the part set up in your indexing table and clamp the "hat" of the tap wrench in the drill chuck (other piloted tap wrenches don't have a "hat" - on those, you grip the pilot loosely in the drill chuck) and slide the pilot of the tap wrench into the "hat". Since the hole is directly under the drill spindle, and the pilot keeps the top of the tap wrench concentric with the spindle, you'll tap it straight every time.