Sometimes an explicit example works better for me than anything else. So here goes.

Consider the sum $S=\sum_{m=0}^\infty m! x^m$. Typically one gets such a series in some perturbative expansion by expanding the integrand in some integral in some small parameter $x$. Otherwise, one carries out Borel resummation to obtain an alternate representation for S. This is a classic asymptotic series. Suppose we are interested in obtaining the sum when $x=x_0$. Consider the partial sums $S_N=\sum_{m=0}^N m! x_0^m$. The error in the truncation in an asymptotic series is given by the next term in the sequence, $(N+1)! \ x_0^{N+1}$ in this case -- we can try to minimise the error by choosing $N$ suitably. Using Stirling's formula for the factorial, and then minimising w.r.t. $N$, one sees that the error is minimum when $N=N^*\sim 1/x_0$ and the error from this $\sim e^{-1/x_0}$. Truncating the asymptotic series at $N^*$ is called the **superasymptotic **truncation by Berry. This simple example teaches us a couple of things: (i) The truncation point gets larger as $x_0$ gets smaller and (ii) The truncation error is non-analytic in $x_0$.

In physical applications, $x_0$, is typically some coupling constant. The truncation point (or the point where perturbation theory in $x_0$ breaks down) could appear at some high order. The non-analytic behaviour in some cases (see comment by Andrew above) are typical of instanton contributions.