+1 Check everywhere to find out where the tight spot is, both front and rear. On a road bike it could be the brake bridge or chainstay width in the rear, likely the brake in the front but this can vary. I have a hybrid where the front derailleur pivot is the limit for the rear tire.

You can use hex L-wrenches as gauges to estimate how much clearance you've got from your existing tires, but be aware that tire sizes are not consistent; some 28 or 32mm (for example) tires are wider/taller than others. They should be more consistent if you stick with the same brand and model and just go with a larger size.