How can I require a generic parameter to be an enum that implements an interface?

Use an ‘&‘ instead:

public class MyWidget<T extends Enum<T> & MyInterface> {
    ...
}

The JLS calls this an “intersection type”, but I can find no mention of it in the Java tutorials. I’ll just say that it does exactly what you were wishing that “extends” would do.

Also, I should mention that you can have as many types as you want in the intersection type. So if you wanted, you could do:

public class MyWidget<T extends Enum<T> & MyInterface & Serializable & Cloneable> {
    ...
}

[Note: this code sample should not be construed as an endorsement of the Cloneable interface; it was merely handy at the time.]