Since I've only been biking seriously for a few years and some of you have been biking longer than I've been alive -- why do I keep getting pinch flats when swapping tires & rims?

Lately I've been swapping a lot of tires and rims around and when I try to re-use a tube I almost always pinch flat it. When I use a brand new tube I don't.

Let's rule some things out:
1) The tires are good. The tubes are good. The rims are good.
2) I don't use levers to put the tire on (but I use them to take them off, usually.)
3) When seating the tire I start at the valve and work my way around to the opposite end
4) I visually inspect the bead to see if the tube is caught between the bead and rim.

Here's my guesses -- either I'm damaging the tube with the levers when I remove the tire; or I'm pinching it when I seat that last little bit of the tire that's hard to get on even though I'm not using levers on that part.

Other than start using baby powder I'm not sure what else to do.

30 degrees, dark and drizzle at 4 am this morning.. still better than no ride at all. Or the trainer.