In the Supreme Court judgement, Chief Justice Dame Sian Elias, Justice Terence Arnold and Justice Susan Glazebrook said they were not satisfied it was necessary in the interests of justice that they heard and determined the appeals.
The justices did not agree that the case raised a question of general or public importance.
"All that this case involves is the application of well-established principles to a particular fact situation."
They also did not accept there was a risk of a substantial miscarriage of justice.
"The Court of Appeal addressed the matters raised by the applicants in detail. We are not persuaded that any material error in the Court's analysis has been identified.
"Finally, the failure of the applicants' business does not make this a matter of general commercial importance.
"The applications for leave to appeal are accordingly dismissed."
When the Balfours took their appeal to the Court of Appeal earlier this year, their lawyer, Eric Forster argued that the $28,000 fine imposed - to be paid to the SPCA - was "unduly harsh".
Mr Forster also expressed concern that TVNZ footage of the raid on the Dannevirke property was excluded from evidence without being viewed in full by Judge Alastair Garland.
Mr Forster also argued that some of the diseases some of the cats suffered, such as ringworm, were not significant and there was "no suffering in the dog population".
The couple kept the animals as part of a breeding operation.
Half of the animals were put down and about 50 cats were treated by a veterinary surgeon. Some of the cats took three years to recover fully.
Crown lawyer Stephanie Edwards said the fine was at the "conservative" end of the available range.
She also defended the length of the animal ban.
The Crown had sought a lifetime ban, but the legislation was not clear as to whether that was lawful, she said.