In a decision released on Tuesday, Justices William Young and Robert Chambers dismissed their application for leave to appeal.
"We are not satisfied that it is necessary in the interests of justice for this court to hear and determine the proposed appeals. None of the matters raised is, when properly analysed, of general or public importance. Nor do we consider a substantial miscarriage of justice may occur if we do not hear the appeals," they said.
Urewera activist Tame Iti, along with Te Rangikawhiria Kemara, Urs Signer and Emily Bailey, were found guilty in March by a jury at the High Court in Auckland of unlawfully possession of firearms during the military style camps in the Urewera Ranges in 2006-07.
However, the jury could not reach a verdict on organised criminal gang charges.
Iti and Kemara were given jail terms of two and a half years- both have since been released on parole, while Signer and Bailey were given home detention sentences.
The Court of Appeal dismissed the quartet's appeal against their convictions and sentences.
In arguing for a Supreme Court hearing, the quartet's lawyers argued the trial judge and crown lawyers had mixed up the admissibility of evidence and directions to the jury.
However, Justices Young and Chambers dismissed their arguments.
