Normally, the Government would agree to the application, when the taxpayer misses the deadline to appeal by so few days. So, there may have been other reasons for the Government lawyers to object to the application.
When you apply to extend time to appeal, you should go to Court if the Government challenges the application. You must explain why you were late and show that you always intended to appeal. Your lawyer can't do that for you. But Ms. Hamilton didn't go to Court. She filed affidavits instead. That should have been acceptable too, if the Government lawyers had a chance to question her on her affidavits and if there hadn't been obvious inconsistencies in the affidavits. Because she didn't go to Court and the affidavits had inconsistencies, Justice Miller refused the application.
Some people assume that the Tax Court judges will always agree to an extension of time. And usually the Court is generous in allowing extensions. But if you are asked to go to Court, be careful if you decide not to go.
Hamilton v. The Queen (2013 TCC)