What if competence has more to do with being able to compete in a professional environment than being excellent at what you do? 
Being able to compete in a professional environment might just have more to do with skills in social engineering and self marketing than with creational skills and accomplishments towards some shared goal.

While excellence seems to be intrinsically driven and an important factor in trying to accomplish something for the greater good, competence is more dependent on external factors and personal benefits. 
To me this feels like the two are not only opposed, but mutually exclusive.

I prefer being surrounded by people who strive to be excellent at what they do which includes sharing their knowledge and skills towards a common cause without their ego getting in the way.

I think we should stop trying to look for competence in people and start looking for excellence and something i would call “collaborance”:

Collaborance — the ability to use your ego for the team, not the other way.

Thomas Schindler