I always remove the tire completely from the rim when I repair a flat as it's the only way to inspect the inside adequately to find what caused the flat in the first place. I'm sure everyone has seen someone do this or done it themselves; you rush to repair the flat by yanking out the old tube, stuff in a new one and inflate it only to have another flat a few hudred yards further along because what caused the first flat is still in the tire.
When I reinstall the tire, I do fit one bead over the rim, then stuff in the lightly inflated tube and finally seat the second bead. Pinch the beads together and check all around the wheel on both sides to be sure the tube is completely inside the tire before you inflate it.