Zach,
If you are not using contact and are treating the connectivity as 'free' where the contacts would otherwise be then you can use modal to identify the culprit.
Method 1: run a constrained model and ask for rigid modes. remember to ask for number of parts x 6 as + 1 as the number of modes. Keep the mesh simple and unrefined. Review the rigid modes
Method 2: (useful if you have contacts) constrain each part fully with it's own constraint. suppress each in turn until the model fails. Use bisection method to home more quickly on the problem if you have a lot of parts/constraints.
Method 3: Review geometry again and pedantically ensure that every component is bonded to something that eventually leads to ground.
To get quick failures use Multi-pass adaptive and limit the maximum edge order to 1. No mesh refinement or anything fancy.
It will be obvious when you find it.
How are you accounting for the axial loads?
Keep us posted