As a nation, we seem hardwired to run ourselves down.  As children, we are taught that it is bad to show off, to brag.  We automatically hone in on the things we could do better.  When I am career coaching, I work with people on their strengths, and it is sometimes like pulling hen's teeth.  Yet once people get started, they discover, to their delight, that they have a wealth of things to be proud of.
Similarly, when I work with people on their 360 degree feedback, their report may be filled with wonderful affirmation about their abilities, but they hone right in on the one comment that hurts.
It's important to remember that 360 degree feedback is about perception, and not fact.  Celebrate the things that define your strength, and work on the areas that you want to grow.
A major retail giant held monthly management meetings and focused on the areas where they were failing. Time and again they looked for ways to improve, but nothing seemed to change the results.  They decided instead to focus on the areas where they were doing well, and analysed what it was that made success. Then they did more of it.  Results soared. 
Try it.