In reading the scroll of 1 Timothy, Paul instructed Timothy, a pastor-teacher, to openly rebuke some elders of the local assembly in Ephesus for misbehaving so that they will NOT continue on sinning. By doing this, he would set things right and prevent other elders from doing the same mistakes. We do not know what the elders did wrong specifically, but it should be bad enough for Paul to write Timothy a letter and to tell him to bring them (the elders who sinned) to the court in front of the assembly for an open rebuke.
Contrary to what many people think, Timothy was not an elder. He was a pastor-teacher who went around with Paul teaching at different places. But in 1 Timothy 1:3 we read that Paul urged him to stay in Ephesus for some time to fix the problem with the elders in their local assembly. He stayed there to teach the believers in Ephesus, along with the elders, how to behave according to the doctrines imposed by the Lord Himself.
1 Timothy 5: 19-21 "Against an elder receive not an accusation, but before two or three witnesses. Them that sin rebuke before all, that others also may fear. I charge thee before God, and the Lord Jesus Christ, and the elect angels, that thou observe these things without preferring one before another, doing nothing by partiality."
Elders are known to be the watch keepers of the flocks of the local assembly. They are the leaders. But remember that they are also human beings like all of us. The elders of today are as susceptible to mistakes as the elders of the early assemblies. Today, we do not have Paul nor Timothy to rebuke the elders who do wrong, so who could dare bring them to court and point out their misconducts whenever they commit one? I believe that Paul gave us a perfect example on what to do in such cases as written in his epistle.
Today, we may not have Paul nor Timothy to rebuke elders and to point out what they have done wrong. But we certainly have the teachings of the New Covenant, particularly the scroll of 1 Timothy, to guide us. Also, we still have pastor-teachers, like Timothy, who travel around different local assemblies and educate us about the teachings of the early apostles as it is written. These pastor-teachers are designated by God as gift to the Assembly (See Ephesians 4:13), not designated by people based on college diplomas. Moreover, these pastor-teachers are full time workers who look out for the good of all local assemblies (in contrary to the pastors of denominations who lead one group). They are the ones who should openly rebuke any misconduct done by elders. And they should do this the same way Timothy did it before: neutral, showing no prejudice, and no spirit of partiality.
As much as we, believers, respect elders, we (elders included) should also respect the pastor-teachers who are tasked by God to keep the local assemblies pure and in line with the sound teaching of the New Covenant. And part of keeping the assembly pure, is dealing with the human errors and misbehaviors done by the elders.