the world is time dependent/transient and nonlinear
if we stay in the static world (time is not important) then your choice of nonlinear vs linear is dependent on your knowledge of the problem
if the displacements are small and the material is linear, then you can run a linear analysis
if you can't safely make either of those assumptions, you need nonlinear
the reason there really is no guideline for this is because everyone's problems are different and because everyone wants a different level of accuracy.
when it comes down to it, every problem should really be run as a nonlinear dynamic problem, but that isn't practical for everyone. that is why we can add assumptions to the problem and use other types of analysis.
in all honesty, if you're getting down to % from straight on the E curve or amount of deflection, you should also realize that the assumptions inherent in your boundary conditions and materials...etc probably account for more error than linear vs nonlinear.
follow the standard guidelines and you should be good
if in question, go to nonlinear, computation time is cheap! and the software isn't too costly. especially considering your problem will run in linear.
if you're concerned, get a couple of standard problems and ask your VAR to do a benchmark. on your end, run some variance tests where you make the appropriate adjustments and run in linear and compare to nonlinear.