The short answer is yes, but...the bird that is referred to as the "Cornish X" or "jumbo cornish rock" is a hybrid. Meaning it wont breed true. So you have to keep more than one seperate breeder flock to hatch them. Another thing is the cornishx is no ordinary hybrid. You can't take just any cornish roo and breed it to a Plymouth Rock hen and presto a jumbo cornishx. It is a multi generational process to produce this bird. The breeder birds used by large corporations like, lets just use Perdue as a possible example, are intellectual property. Compare it to any patented technology the birds with the recipe for a jumbo chicken are a closely guarded secret. These companies have unlocked natures secret mixture of genes and in so doing spent of millions of dollars and countless hours in research. So they don't let just anyone have the birds with those prized genes. They will sell them for the right price though I'm sure. Even most farms that produce our supermarket meat birds don't have the breeders on the farm. They are mainly contracted growers of chicks they never bred or hatched.
So you CAN keep and breed these birds. It is just practically impossible formost.