I'd take the bike to the shop you intend to buy the tyres from.
When I bought my Cannondale, the guy in the shop mounted another set of tyres because I wanted bigger than the standard set. (unfortunately I can't remember the size but for some reason I'm thinking 32mm) A couple of days later I noticed the tyres weren't at their max pressure and I pumped them up. The tyre was stuck against the frame. I went back to the shop and he happily swapped them for 28mm. Rim size may not be your only problem.