Simple tester checks Christmas-tree lights article gives you ideas how to test mains powered Christmas lights. The article presents a simple test circuit. Some more testing ideas can be found at Christmas Lights and How to Fix Them web page.
LED lights are becoming more and more popular for common in Christmas lights for many good reasons. They consume considerably less energy than traditional light bulbs and fail much less often. If you LED lights fail, you can find ideas how to fix then at LED Christmas Lights and How to Fix Them web page.

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Nonfailing light-bulb-string story causes readers to think
http://www.edn.com/electronics-blogs/other/4359581/Nonfailing-light-bulb-string-story-causes-readers-to-think
Our Leading Edge item on how series-wired Christmas lights self-heal prompted some interesting reader responses suggesting alternative ways that a failing bulb does not cause an entire string of bulbs to fail (”Simple scheme saves string-light situation,” EDN, Dec 23, 1999, pg 24). Apparently, some light strings use another mechanism, and we have some hands-on evidence to confirm this fact. This approach involves no software, no microprocessors, no logic gates, no nanosecond timing—just some passive light bulbs and clever materials engineering.
The way I see it working is this: The two support wires have a coating on them as you suggest, but it is not resistive; it is insulating but thin. Around the two filament wires is wound a number of turns of uninsulated fine wire. As long as the string of bulbs is working, each bulb has 2 or 3V across it, and all is well. When a filament burns out, no current flows in the series string, there are no voltage drops, and the full 110V appears across that open bulb. This ruptures the thin insulation, and the two filament posts short out through the wire and spot weld the whole thing into a short.
Measurements seem to bear this theory out. Once in a while, the spot welding doesn’t work, and, when the bulb goes, the whole string goes out, due to the insulating layer still working.
As each one burns out, a slightly higher voltage is placed across each of the other bulbs, reducing their life. As further ones burn out, this effect accelerates, and you can see a whole string go in a couple minutes—the later ones like flash bulbs.
Until, of course, the resulting higher current fuses something in the circuit and breaks the circuit.
Because five replacement bulbs are sold for about the same price as a whole new string, there is no real point in trying to repair one of these chain-reaction burnouts.
Moral: Replace your burnt bulbs as soon as possible—just like they tell you to in the instructions
I really can’t wait, the defending Quiz champions will be returning this year.