The issue with the N-body (N > 2) problem is that it is sensitive to initial conditions. In other words, a chaotic system.
No matter how accurate your approximate solutions are, if you don’t have highly accurate initial conditions, you can have wildly varying results in a relatively short period of time.
Depends on what you mean by relatively short period of time. The end state of the solar system is unpredictable. It is possible that Jupiter will perturb Mercury’s orbit so that it crosses earth’s. However on a mere millennial scale, given the current knowledge, quite predictable.