People can fully know themselves usually after having undergone a tremendous change in life or finally accomplishing all their goals. Some monks and other driven people will mediate day after day and contemplate the meaning of life and who they are to better understand the world around us. It is possible to fully know ourselves.
People cannot fully know themselves because they have never seen themselves from a different point of view. People are always thinking and our perception is constantly changing. Everything changes and so do we. I believe that it is impossible. Scientist say that if we even saw ourselves in person we would not recognize our own self. So I do not think that people can ever fully know themselves.