Do we ever need to prefer constructors over static factory methods? If so, when?

Question

I have been reading Effective Java by Joshua Bloch and so far it really lives up to its reputation. The very first item makes a convincing case for

Answer 1

Constructors are boring, you know exactly where to search for them, you know the scope of them, they are somehow predictable. They are there to construct the object itself, not to do god-like jobs. If you are doing too much other stuff that has not much to do with constructing the object you know you are doing something wrong. And it is not possible to "heal" this just by renaming the method. So they tend to be more focused, "better" in code quality when you have not top-of-the-line developers. (When you have the latter the points I mentioned don't matter.)

Boring stuff. Let's do something more fun and intellectual challenging...

Just for the records: I don't neglect the points mentioned in the question and other answers, I just wanted to add this additional point.