tail probabilities for the sum of independent Laplace random variables

How might I find tail probabilities (pr X>x), or a reasonable approximation, for a variable that is the sum of independent Laplace random variables?

I'd suggest using the characteristic funciton of the Laplace Distribution. The characteristic function of the sum of N independent Laplace r.v.'s is the product of their characteristic functions. Then, you can invert the characteristic function to get the density function of the sum. Integrate this result and subtract from 1 to get what you want.

If you don't feel like getting the exact value, you can use Cantelli's Inequality to estimate the tail probability. If $X_i\sim$ Laplace($\mu_i \, b_i)$, and $Y=\sum X_i$, then you can apply Cantelli's inequality by setting $\mu_Y=\sum \mu_i$ and $\sigma_Y^2=\sum 2b_i^2$ and $\lambda = x$ to get:

$P(Y-\mu_Y\geq x)\leq \frac{\sigma_Y^2}{\sigma_Y^2+x^2}$ if $x\geq0$ OR

$P(Y-\mu_Y\geq x)\geq 1-\frac{\sigma_Y^2}{\sigma_Y^2+x^2}$ if $x<0$