Recovering a Differential Equation From the Transfer Function of a Cascaded System With respect to the below discussion, consider that we are talking about LTIC systems characterized by constant coefficient ODEs.

Consider a cascaded system whose transfer function H(s) is given by 

$$H(s) = \frac {s+2}{(s+2)(s+3)(s+1)} =\frac 1 {(s+1)(s+3)}$$

"Recovering" the defining differential equation from this transfer function yields 

$$(D^2 + 4D + 3)y = P(D)x$$

where $P(D)$ is unknown but irrelevant for the purposes of this discussion.
What are the characteristic modes of the system?

Is it acceptable to simply say that the natural response $y_n$ is given as below?

$$y_n = Ae^{-t} + Be^{-3t}$$

It would seem to me that the answer is no. Why can we recover an ODE from a transfer function in general. For instance, what if the cascaded system had've had $(s-2)$ factors rather than $(s+2)$ factors that cancelled (ie. one of the subsystems had a pole at $s=2$). In such a case, would it not be wildly incorrect to recover the ODE above since then we are ignoring an internal characteristic mode that blows up $(e^{2t})$?
You have discovered pole-zero cancellation, which is one of the reasons that state-space systems description was invented.  Another reason is because when you go from a differential equation to a transfer function, you lose information about the structure of a system that you can choose to preserve in a state-space description.

In general, you should be very careful when you see a pole-zero cancellation, because even if the pole is stable, it can still affect the behavior of the system on startup, if there is a transient that pushes the system into nonlinear behavior and excites the extra pole, or if the pole and the zero don't actually match up perfectly in the real world.

And yes, if you have an unstable pole that's cancelled by a zero then Bad Things will happen to the system that simply aren't described by the transfer function.