When is an input signal considered small for BJT and JFET devices?
It's not that you have to use small signal modelling when the signal is small...it's that you can use small signal modelling when the signal is small. A small signal model will typically be much simpler to compute than a large signal model.

The reason the small signal model is simpler to compute, is that it linearizes all the circuit behavior around an operating point. It assumes that any change in the input signal will be reflected by a proportional change in the output signal. Equivalently, it assumes that with sinusoidal inputs, there will be no harmonic frequencies generated at any of the circuit nodes. In the small signal regime, the simulation can be done using pure linear algebra, without having to solve transcendental functions or do numerical integrations. 

So when are you allowed to use a small signal model? Basically when the signals are small enough that the nonlinear behavior is negligible. How small this is depends on the exact details of the devices and the circuit, and on how accurate you need the result to be. As the signal amplitudes increase, the small signal model will be more and more inaccurate. It's up to you to decide when this inaccuracy is too much.