it depends on your state, and what car is in question. Old street rods can legally run open-wheeled, because when they were originally built, it was perfectly legal to run without fenders. it's what's known as a "Grandfather clause" or in other wors, because it was legal when it was first made, it is exempt from future changes in the law.
Now, on all cars from the 50's & up, they have to have fenders in tact, and the wheel cannot protrude beyond the fender lip. This, however, did not stop the resourceful tuners out there.
Street rodders figured out that, to run large drag slicks in back, they could "Tub" the back end, or in other words, chop out the rear wheel wells, and replace them with deeper, aftermarket wheel tubs, and installing a slim rear end. the result is a car that wields extremely tall and wide tires in back, but remained legal by tucking them neatly under the car.
Later generations also found that by changing to a different wheel offset they could run wider tires as well, others eventually came up with "Fender Flares", which extended the lines of the fenders to accomidate the fatter rims and tires.
While, in most cases, it's illegal to run a tire beyond the sheetmetal boundaries, there are ways around it to keep your ride legal and get the added performance benifits of a wider-than-stock wheel/tire combo.
Also, you can run a wider tire easily, the fender has enough free space to accomidate it. the tire cannot protrude beyond the widest point of the fender, beyond that, you either gotta go deep dish, or run flares to keep it legal.
BTW< this law is in place because, without fenders to cover the tires, the tires can kick up rocks and debris to the point that everybody is constantly getting cracked windshields, dents, etc from the guy in front of them. it became almost epidemic in the early hot-rodding days, so Johnny Q. Law stepped in and said, "No more from this point on!"
Some vehichles, like vintage Deuce street rods, Horse-drawn carts, tractors and motorcycles, can get away with running open wheeled, the rest of us are just plain SOL.