John 6:64, "But there are some of you that believe not. For Jesus knew from the beginning who they were that believed not, and who should betray him."
Jesus, our Example for how we should live our lives, knew from the beginning who, in our terms 'He could and could not trust;' yet, He served them anyway without reservation. Why? Jesus gives us the answer in John 9:4, where He said, "I must work the works of him that sent me...."
You see, Jesus trusted and obeyed God the Father. We are told in Proverbs 3:5 to "Trust in the LORD with all thine heart;..." Notice that it says "all" thine heart. Just like in the example of Jesus, ALL of our trust and obedience belongs to God - even when that means loving, serving, working alongside, and washing the feet of the untrustworthy Judases in our lives.
When all of our trust belongs to God, He will guide and direct each step of our lives' path for our good and in obedience to His will (Proverbs 3:6-7). The question then is not, "Can I trust my friend / co-worker / child / spouse / etc?", but "How can I trust God and show it by serving others?"