The trick is to get the side of the tire that's opposite the last few inches of bead into the lower center section of the rim. It actually works best if you fit the section adjacent to the valve stem last.

1. Pinch the beads together so they don't seat on the rim.
2. Hold the wheel vertically in front of you with the valve stem down.
3. Grasp the tire at 3:00 and 9:00 and push downward. Try to concentrate all of the slack where you need it.

Most times, using this process, I'm able to install and remove tires without using tire levers. I think that most of the difference is in the rim. Some rims, like Rhyno Lites, are wide and shallow and are difficult to install tires onto.