http://www.TireRack.com says your Corolla came originally equipped with the P185/65R-15 size.
http://www.tirerack.com/tires/Selec...toModel=Corolla+CE&autoYear=2003&autoModClar=
To verify, locate the tire placard for your car. This lists the original equipment size that was installed on your car at the factory. The tire placard is a small sticker that is usually located on the end of the drivers door, (but it can be on any of the other doors too), on the backside of the fuel door, on the underside of the trunk lid or on the back of the glovebox door.
Yes, the larger tire could be installed, but the drawback is the speedometer and odometer will no longer read correctly and because a larger tire has more drag, the fuel economy of the car will be reduced. Also, with a larger/taller tire, the cars center of gravity is raised and it will lean more in turns, not handling as well.
Tire Rack has almost 50 tires in the correct P185/65R-15 size for your car -
http://www.tirerack.com/tires/TireSearchResults.jsp?tireIndex=0&autoMake=Toyota&autoYear=2003&autoModel=Corolla+CE&autoModClar=&width=185%2F&ratio=65&diameter=15&sortCode=37550&skipOver=true&minSpeedRating=S&minLoadRating=S&tab=All