Rim width makes a difference too. I was running 2.25 tires on a Schwinn middleweight with standard rims. It was a tight fit but they worked. Then I switched to a heavy duty rim which was wider, and the same tires made contact with the frame, the rear forks specifically. I put 1.95 tires on with the heavy duty rims and they fit like the original 2.25's.
I guess if the rim is smaller, the tires bend up a little bit. With a wider rim they're free to expand laterally.