Yes, as a couple of people have said, the springing medium is RUBBER, not HYDRAULIC FLUID (actually mostly water in this case), as someone has incorrectly said, which is incompressible and only provides damping and fore and aft coupling. When the rubber is hard, as it will be, there is basically no springing, and the stresses on the rest of the car are increased. Once they get above the fatigue threshold in some part or other, you are well on the way to a structural failure. It may be the top arm, which may fracture near the knuckle joint, or maybe the subframe or bulkhead structure, or the welds between side members and front cross member of the rear subframe, or maybe somewhere else altogether, depending on the overall condition of the car.
But any car, or any sort, will break due to fatigue fracture after a while, if the suspension is too hard. Some, not Minis, and often much more expensive, have been known to break when there is nothing wrong with the suspension, due to ignorance on the part of the designer, who allowed the peak repetitive load on some part or other to be above the fatigue threshold. Some, apparently, were made in Bavaria, which was inexcusable on any car, especially an expensive one, in an age when computing facilities of which Issigonis could not even have imagined had become commonplace, and should have detected such gross errors. Issigonis got rid of all that once and for all by thrashing the Orange Box prototypes for indecent miles on horrendous surfaces, which led to the solid mounted subframes being introduced. Then someone reintroduced fatigue problems along with the abominable rubber mounted front subframe, but I digress....
Actually the amount of rubber in the spring in the displacer is fairly similar to that in the dry suspension cone, which should not be surprising, as you need a certain volume of rubber to do a certain amount of work. Less rubber, same amount of road roughness, and it will overheat. And, it is more or less in the form of a cone, so it acts in shear, rather than compression, same as the cones.