Legal Question in Criminal Law in Michigan

It's hard to answer without knowing more of the facts. There are a number of factors that a judge must consider and strictly follow when taking a plea. It could be any number of things: the defendant's admission didn't meet one or more elements of the offense, perhaps he believed the defendant was not making this decision freely and voluntarily, maybe the judge felt like he couldn't go along with the sentencing recommendation... Could be any number of issues. Usually the judge will say why they can't accept the plea on the record.

Or is what you are saying is that a judge refused a particular plea offer?

The most common reason is one that Jared mentioned: the defendant's factual basis (what he/she is telling the judge he/she did that makes him/her guilty) did not cover all the elements of the crime ... or the defendant's "I'm guilty..." words were undermined by other statements that tell the judge the defendant doesn't think he's really guilty, or is invoking a legal defense to the crime. A common example is a plea to an assaultive crime and the defendant also says that he did the assaultive act "in self defense". Some buzz words that raise this issue are "I was defending myself" or "the victim hit me first". Judges should only accept guilty pleas from people who are truly guilty, not people who are simply trying to get the case done with. Jared mentioned other less-common reasons why judges might reject a plea.