My wheels are [this size]. How big of a tire can I fit on them?

According to Paul Bratina, rim width should be 70-100% of the tread width. That is, the wheels are generally narrower than the tires fit to them. The optimal ratio is to have the rim about 90% as wide as the tire.

Fitting tires that are too wide for the wheels will pinch the tires too much, resulting in the tire having a poor shape. It might also slip off of the wheel, not to mention being almost impossible for the poor tire tech to get mounted. Narrow tires on wider wheels will tend to slip around. For more information, consult a good tire shop.