I weigh around 215 lbs and my bike adds about 35 lbs more. Call it 250 lbs. I have ridden the entire Katy Trail 8 times on 700x32 tires.
However this past June it seemed like the trail was rougher than in the past. They seem to be using pea-gravel for repairs instead of limestone. I bought a pair of 700x35 tires which is as big as will fit my bike with fenders on it. I figure a slightly larger tire and about 10 psi less pressure will make for a better ride. I haven't tested this out yet on the Katy.