Restoration of the term for appealing to the court due to illness

I don't quite understand in which specific cases my illness can be a valid reason for me to have the terms for appealing to the administrative court restored. If I was on sick leave for a long time and saw doctors, but was not in a serious condition throughout this entire period, can they refuse to restore the term for me? What evidence needs to be provided for the court to genuinely recognize my illness as a valid reason for missing the deadline for filing a claim?