Dear In├®s There are many reasons why a solution will fail to reach the end time. Sometimes an error occurs, such as the "highly distorted element" error. This is resolved with better element shapes. Let's exclude that from the question you ask.
There are Static Structural models where the solver fails to find equilibrium when increasing the load by a small increment. A common scenario is in structures that have a snap-through behavior. The normal load incrementing method fails, which is what you were experiencing. If the load could get past the instability, there is a statically stable configuration on the other side of the instability, but the normal algorithm can't get there. That is what turning on stabilization does, it introduces artificial forces to stabilize the structure to get it past the instability to get to the other side.
There are other technologies that can be helpful with hyperelastic materials that reduce the difficulty the solver has in finding equilibrium. One of those is called the mixed u-P element formulation. Without that, the deformation of the nodes determines the pressure in the element. That is normally fine because the material is compressible and small changes in nodal deformation creates small changes in element pressure. But hyperelastic material can be incompressible or nearly so with Poisson's ratios of 0.49 or higher. Now small changes in nodal deformation creates huge changes in element pressure, which makes find equilibrium very difficult. The mixed u-P element formulation adds a pressure degree of freedom to the element making it much easier to find a set of nodal deformations that have equilibrium. You should definitely be using this in your model.
Best regards Peter