Where it is thin is the real major point. Out near the muzzle is much less troubling than near where I stand and hold the gun. I know of one gun that passed proof with .017 barrels not long ago. That was near the muzzle not near the chamber end. When you see just "passed proof", do not assume that that means the barrels are robust, just that they did not fail when proofed. Like I mentioned earlier thin barrels are very easy to dent. If you are looking for a shooter I would be very leery of any gun with barrels thinner than .025. If you are looking a collector type gun, which will be rarely if ever shot, then thin should not be that big of a deal. Just handle them gentle.
