[rolldat777]

I think a lot of the answer is 'chaos'. Most nontrivial systems are chaotic (generically, iirc more than 3 unconstrained d.o.f. + non-linearity) and most chaotic systems quickly erase knowledge of the initial conditions (https://en.wikipedia.org/wiki/Lyapunov_exponent). That being said, within chaos, there are fractal regions that survive pertubations (https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%8...) and this is indeed related to why the solar system is more stable and predictable (can read about KAM theorem and Jupiter).

In the bouncing ball example, there are periodic orbits that are created for some balls, in which case you can perfectly predict their past and future. But a lot of measurements we do make about the world tend to be statistical measurements that don't depend on measuring the precise trajectory of individual particles over long periods of time. So these issues tend to not be important unless precise details are needed.