Let me start by saying that by "being good" I do not mean by the standards of Mark 10:18, which would guarantee an answer of "no". Nor am I asking if they still bear the image of God (though no longer the likeness) and are valued by God; this question would guarantee an answer of "yes". I mean, can they behave well by human standards? Oh, and just because some choose not to does not mean they could not have.
It's worth adding that even by human standards, we are a species of stinkers. Nothing drives this home so forcefully as a daily examination of conscience. As Hilaire Belloc said, we
... pretty nearly all day longThe ancient Greeks knew this, which is why we have the story of Diogenes carrying a lit lamp during the day to look for an honest man. If the question is to be interesting, we can't set the bar very high.
Are doing something rather wrong.
We do need to set it high enough, though, that "being good" is inconvenient. It is only through some degree of sacrifice that we can be sure there is real commitment.
Now imagine that you have a friend who insists, loudly and frequently, that Napoleon never existed at all, being entirely a fiction created by the British and Russian ruling classes to keep their populations awed and subjugated. In spite of that, this friend insists on dressing and acting like Napoleon. He persists in this behavior even though it has cost him friends and job opportunities. If anyone says he does not really look like Napoleon, he becomes angry -- almost as angry as he becomes if anyone says there was a real Napoleon for him to look like.
This behavior is possible, but by no means rational. Furthermore, when you say the friend "looks like Napoleon", you mean there was a real man whose appearance can be accurately determined (in spite of conflicting accounts) by careful research; all the friend can mean is that he looks like the popular image of Napoleon.
The point is, of course, that anyone who loudly insists that there is no real moral standard, but who nevertheless behaves as though it did, is not really behaving rationally, and he certainly is no judge of what good behavior really is. Nevertheless, he may be better than his beliefs; many people are saved from being monsters by being slightly irrational.