Absolutely. A lot of my thesis could be considered classical mechanics (orbital dynamics, specifically). A very old problem that is still open is the question of whether the solar system is stable. (Newton worked on this question 300 years ago.) The state of the art can only reliably integrate orbits for the next few million years. After that, numerical limitations and chaos make it hard to predict exactly what the dynamics in the far future will be. All we can do is make statistical predictions (which we hope are unbiased).

The best predictions are that there is a ~1% chance that two planets will collide before the Sun dies. Here's a nice article:

http://www.scholarpedia.org/article/Stability_of_the_solar_s...

Does this count as having testable solutions?

I remember being told once that we don't have good simulation techniques for predicting the sounds produced by, for instance, the collision of two solid bodies. Lots of work has been done on modeling the kinematics and the optical appearance, but not on the audio side.

I don't know if this is still true.

Work has been done on this:

https://www.youtube.com/watch?v=flOevlA9RyQ

https://www.youtube.com/watch?v=ZaFqvM1IsP8

3-body problem is still unsolved if collisions are involved

there's still interesting issues of numerical accuracy in the point-mass case: E.G. is there a "good" way to calculate positions to arbitrary precision given starting conditions given in arbitrary precision, and what the relationship is.

I think the granular pile problem is still unsolved: Can you predict the cone angle of a pile of granular material given a small set of initial parameters?

if you want to include fluid mechanics: https://en.wikipedia.org/wiki/Turbulence https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...

I heard somewhere that when we apply all our knowledge to compute the position of the moon, it is a couple of feet off when we measure it, and we don't know why.

I have been trying to find a source for that, but I've failed. Perhaps someone reading this knows? :)