I just bought a new refrigerator and have connected my previous copper water line to the appliance. It seemed leak-free, but about an hour after we pushed the fridge back in place we noticed a very tiny puddle so I pulled it out and started over this time using pipe threading (??) and thought that worked. It once again seemed okay so I pushed it back in place again....wrong. It still had a drop or two showing up. Since I've pulled it out from the wall, it looks dry again. Did I tighten it too much? Did I not use the white tape properly (I only wrapped it once and this is my first time using the stuff)? Can I use more of it or something else? Any suggestions would help. Thanks.
Your question doesn't mention where you're located, but I'll speculate USA and from that infer that you're working with 1/4" tubing and a compression fitting -- this is the most common arrangement for refrigerator water supply in the US.
Here is a photo to give names to the parts you're probably working with. I didn't have any copper tube on hand; the blue tube is polyethylene plastic instead. It's shown with a white plastic (Delrin/acetal) ferrule and a brass insert. Your copper tube should have a brass ferrule like the one shown loose in the photo. The brass insert is used only with plastic tubing.
The compression fitting is designed to seal dry, ie without the white Teflon tape or pipe joint compound. It can leak if there's debris in the joint, if it's not assembled square, if a part (usually the ferrule) is nicked, or if it's overtightened.
If a leaky compression fitting can't be resolved by cleaning the mating faces and carefully re-assembling, usually the next step is to cut off that last inch or so of the tube, slide a new ferrule onto the tube, and assemble again. Plastic tube can be cut with a knife but your copper tube should be cut with a proper tubing cutter.
Some say that wiping a bit of pipe joint compound on the ferrule can help seal a small leak. Maybe it can; I haven't tried it -- it seems a bit of a hack and I've always preferred to trim the tube and install a new ferrule instead.