The premise is a good one...

Who wouldn't want a higher level of emotional intelligence? Studies have shown that a high emotional quotient (or EQ) boosts career success, entrepreneurial potential, leadership talent, health, relationship satisfaction, humor, and happiness. It is also the best antidote to work stress and it matters in every job — because all jobs involve dealing with people, and people with higher EQ are more rewarding to deal with.

Most coaching interventions try to enhance some aspect of EQ, usually under the name of social, interpersonal, or soft skills training. The underlying reasoning is that, whereas IQ is very hard to change, EQ can increase with deliberate practice and training.

...but you can only improve if you get accurate feedback.

And some people are more coachable than others.

I have to agree that "coachability" is a key factor for someone to develop emotional intelligence. I have seen this to be true in my own life as well from spending years helping others grow in their EQ. A person can have many great qualities, but if they don't want honest feedback, then any development is pretty much stopped dead in its tracks. The conundrum is that those with lower EQ are the same ones who have more difficulty with feedback...thus, it can be a vicious cycle. As the HBR article says, "There is an old joke about how many psychologists it takes to change a light bulb. Just one — so long as the light bulb wants to change." For people to change, I have found that they need not only the desire to change but also a sense of trust in whomever is giving them the honest feedback. Desire and trust coupled together are powerful forces...and, unfortunately, in short supply. So for the light bulb to change, it needs to want to change, but it also needs to trust the person helping them accomplish it!

Yes, I think that's really brilliant that you've pointed that out.  I think that I am one of those people that is "coachable"--- IF I trust the coach and we can be in the dialogue together.