I just found this out by accident
When [Character] is immune to [specific status] ( doesn't matter if it's effect materia or accessory or armor )
and something inflicts [specific] status on [character] then that status can never be cured unless the removal method, Item or Attack, also breaks status defense.
An example:
Cloud and Tifa fight Enemy A. Both are immune to silence.
Enemy A first inflicts silence on Tifa. This silence cannot be healed by regular methods. It even carries over through death. killing Tifa and resurecting her will still leave her with silence. Only an attack or item that breaks status defense and removes silence ( or similarly, an attack that causes silence, breaks status defense but is absorbed by Tifa via element ) can successfully remove it.
I haven't tested if status breaking alliments can be cured if the character is not immune.