Are there any non-Jahn-Teller type distortions?
Apparently there is some theorem e.g. to be found in an article from Isaak B Bersuker with title "SPONTANEOUS SYMMETRY BREAKING IN MATTER INDUCED BY DEGENERACIES AND PSEUDODEGENERACIES" in Advances in Chemical Physics, Volume 160, First Edition. Stuart A. Rice and Aaron R. Dinner. © 2016 John Wiley &amp; Sons, Inc. Published 2016 by John Wiley &amp; Sons, Inc; p 187-

The theorem reads like: 

"DEGENERACY AND PSEUDODEGENERACY ARE THE ONLY SOURCE OF SPONTANEOUS SYMMETRY BREAKING (SSB) IN ATOMIC SYSTEMS"

And can be reformulated by:

" [...] can we get structural instability of high‐symmetry configurations in any electronic state without involving interaction with other states [...]? The answer to these questions is no: any instability and SSB in polyatomic systems leading to their formation, deformation, and transformation involves at least two electronic states, degenerate or pseudodegenerate. This is the two‐state paradigm in structural instability."

So in essence that means all types of molecular distortions can be traced back to some JT effect.