You certainly can write your own class with a "public void run()" method in it. This class would not implement the "Runnable" interface. However, what you cannot do is then use this class to create a Thread. Even though the class has the "run" method defined, when you create a Thread with a Runnable, it is expecting something of type "Runnable," as this is the only way to gaurentee that the "run" method exists in the class.
For example, suppose you mistyped "run" as "ryn". Without implementing Runnable, the compiler would compile happily. However, with Runnable, you would get an error message stating something to the effect of "Your class is incomplete -- you need to implement the "run" method." You look back, thinking But I did implement the run method and discover that it is misspelled. You correct the misspelling and everything compiles happily.
(No, this is not far-fetched. Someone here recently had this problem, albeit with the actionPerformed method in another interface -- which was spelled actionperformed -- without the capital 'P.')
Piscis Babelis est parvus, flavus, et hiridicus, et est probabiliter insolitissima raritas in toto mundo.
As Joel said, there is nothing stopping you from creating a run() method without implementing Runnable. In fact, there is nothing special in and of itself about the run() method defined in Runnable. What makes it special is that the Thread class expects to find this method in order to call it. You can create a Thread object that uses another class that implements Runnable and the expected run() method. For more details about how this works, you should google or use the Saloon's search tool for something about threading in Java.