I often see the people who refuse a responsibility. The reason is usually referred to: Lact of capability, shame, fears, another is better, etc.
The rejection is based on willingness to be changed. It is better feeling to stay in current condition. Fail to see something valuable behind the responsibility has been asked to do.
If we feel lack of capability, this is the time to be capable. Learn to dare, dare to learn and learn something new. If you have a responsibility asked for in something where it surely can make you grow and mature, why not saying "why not me" instead of "why me".
Of course you have to be wise for choosing which responsibility will allows you to change better. This is not easy, but take the challenge for you. Be stronger, more capable in every assignment given to us. And you may believe SOMEONE unseen will carry you on. Go ahead, why not me?