I run 700x28's on road bike wheels on my surly no problem and 700x32's should also fit on the rims...

However, if there's not enough room between the tubes of the bike & the rims, you can't fit a wider tire on there. That mixte looks like it'd be fine, because of all the room between the back tire & the seat tube - some of the modern bikes don't leave much room for bigger tires.