Is there a verse in the Bible that says, where the Bible speaks, let him speak, where the Bible is silent let him be silent? Does the church of Christ really hold to that teaching? Are some of the things that the Bible teaches ignored by the church members and do the church members address too many things that are not mentioned in Gods word? Maybe I should say church leaders instead of members.
While the quote, “We speak where the Bible speaks, and are silent where the Bible is silent,” is not from the Bible, the principles expressed are. “If any man speak, let him speak as the oracles of God.” (1 Peter 4:11) “Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.” (Colossians 2:8)
I will admit that individuals, and even congregations, may at times violate these principles. In my experience it is more often a case of binding something which the Bible does not bind, rather than not doing what the Bible specifically says to do. For instance, I have been in congregations where some people considered smoking, drinking, dancing, or men and women swimming together as sins, although the Bible does not say anything about them. There is such a history of other groups leaving out things the Bible specifically commands, such as immersion of believers for forgiveness of sins, that most tend to keep to the strict obedience to those things specifically commanded.
The area where the most disagreement occurs is the authority of the silence of the scriptures. That is, if the Bible commands one thing very specifically, can a person add something else to that which is not included in the specific command? The most noticeable example of this is the question over the use of musical instruments in the assembly. One group says that since the Bible specifies “sing and make melody in your hearts” that excludes any instrument other than the human voice. The other side argues that since God did not say “thou shalt not use mechanical instruments of music in the assembly” such instruments are allowed. There grows up a big controversy over which commands are specific and which are generic. (I choose to argue that question from history rather than the silence of the scriptures. It is much stronger to argue that the early church specifically chose not to use instruments for 500 years than it is to argue whether the Bible commands against them by its silence.)
Those of us who believe in the power of the silence of scripture have to be careful. Those arguments can only be used when there is a specific command to do something. That is, when the Bible says “repent and be baptized” we can say that somebody would be wrong to say “repent and be baptized and sign over all your worldly possessions to the church” in order to be saved. On the other hand, when there is no specific command about something, the silence of the scriptures becomes permissive rather than restrictive.
There is a lot more I could say, but I have given the basic answer to your question. Others have gone into detail much better than I would choose to for this site.