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I wore such a breastplate for a few months last year and got skewered halfway to death before I realized that although I was wearing a breastplate of righteousness, it was not the one prescribed by God as part of his armor. That breastplate I had on was based on my own righteousness, not God's!
A breastplate of our own righteousness is strong to look at, but is inherently weak because it is based largely on our own verdict of ourselves, or the verdict of others, and both are fickle. As long as we judge ourselves (or are judged) favorably, the breastplate we wear appears strong and fills us with confidence. The moment this changes, the breastplate splinters and we become vulnerable to attack from the enemy.
Paul realized this early in his Christian life. In his days as a Pharisee, Paul had it all. He was brought up in Jerusalem as a disciple of Gamaliel and was a very highly respected member of the elite Sanhedrin. He was fanatical about the law, and even among the Pharisees who were renowned for their strict adherance to it, Paul was in a league of his own. He was about as righteous as they came. Yet, when he discovered the righteousness that comes from God, he considered everything else "rubbish".
What is more, I consider everything a loss compared to the surpassing greatness of knowing Christ Jesus my Lord, for whose sake I have lost all things. I consider them rubbish, that I may gain Christ and be found in him, not having a righteousness of my own that comes from the law, but that which is through faith in Christ--the righteousness that comes from God and is by faith. (Philippians 3:8-9)
The breastplate was made from strips of iron laid in an overlapping fashion, secured by leather straps. When these strips are attached in such a fashion, it makes for an extremely strong, yet flexible defense against enemy swords and stabbing weapons. With this flexibility the soldier could move freely and use his shield and sword without the breastplate getting in his way.