In Java, is there a legitimate reason to call a non-final method from a class constructor?

There are times it can be very hard not to.

Take Joda Time, for example. Its Chronology type hierarchy is very deep, but the abstract AssembledChronology class is based on the idea that you assemble a bunch of "fields" (month-of-year etc). There's a non-final method, assembleFields, which is called during the constructor, in order to assemble the fields for that instance.

They can't be passed up the constructor chain, because some of the fields need to refer back to the chronology which creates them, later on - and you can't use this in a chained constructor argument.

I've gone to nasty lengths in Noda Time to avoid it actually being a virtual method call - but it's something remarkably similar, to be honest.

It's a good idea to avoid this sort of thing if you possibly can... but sometimes it's a real pain in the neck to do so, especially if you want your type to be immutable after construction.

One example is the non-final (and package-private) method HashMap#init(), an empty method which is in place for the exact purpose of being overriden by subclasses:

/**
 * Initialization hook for subclasses. This method is called
 * in all constructors and pseudo-constructors (clone, readObject)
 * after HashMap has been initialized but before any entries have
 * been inserted.  (In the absence of this method, readObject would
 * require explicit knowledge of subclasses.)
 */
void init() {
}

(from the HashMap source)

I don't have any examples of how it's used by subclasses - if anyone does, feel free to edit my answer.

EDIT: To respond to @John B's comment, I'm not saying it must be good design since it's used in the source. I just wanted to point out an example. I do notice that each HashMap constructor takes care to call init() last, but this is of course still before the subclass constructor. So an amount of responsibility is falling to the subclass implementation not to muck things up.