Applications for Green Card Renewal/Replacement may be denied for various reasons. Two of the most common reasons would be criminal activity and adjustment of immigrant category.
Criminal activity means that you committed a crime and are therefore no longer seen as a valuable asset to American society. However, this is really only applicable to major crimes such as felonies. Minor crimes such as speeding tickets would probably not result in a denial of a Green Card Renewal/Replacement application.
Adjustment of immigrant category means that your circumstance has changed in such a way that the reason you were granted a green card is no longer applicable. For example, if you were granted a Green Card through marriage to a US citizen but have since gotten divorced, your application for a Green Card may be denied.
If your Green Card Renewal/Replacement application was denied, you do have an opportunity to ask USCIS to rethink their decision. You may submit a motion to reopen or a motion to reconsider with the same office that denied your application. This will cause USCIS to reexamine your case and reconsider their decision.
In the motion to reopen, be sure to clearly state and provide evidence for why your application was wrongly denied and why USCIS should reconsider.