Awhile back I was changing my bike tire in the warmth of my garage, with all my useful tools available and a good bike pump, and yet I still continually failed. I kept pinching the inner tube into the tire and popping it when I would put air into it. After going through all 3 of the inner tubes I had at home, I took my tire to a local bike shop for assistance. They sent me home with a nifty tool that helps to pull the tire onto the wheel (as you are nearing the end and it is getting very difficult). Because of its angle it helps to avoid the pinch. I am sure it is not fool proof, but a worthwhile investment in my mind (about $10), and I have not popped a tire since. This appears to be the one I have and it is called a tire bead jack.
Here is a tool I found recently that has helped me fix a tire without popping the inner tube from getting it punched between the wheel and the tire.