The old myth that if one lamp blows, the whole lot will go out, is old. For a long time now, the lamps have had a "secondary" link which completes the circuit when the filament breaks, thus making the broken lamp easy to find.
Clearly a concern here was that after so many lamps blow, because they are in series, the voltage and current across the individual lamps will become too great, and they will all start blowing quicker until there are no lamps left. To combat this, fuse lamps were developed. Fuse lamps have a slightly higher-rated filament than normal lamps, and will blow when too much current tries to flow. They do not have a secondary link.
Though I did call into question the effectiveness of this system a few years ago.We bought a new set of lights, of this type from B&Q about 3 years ago. On getting the set home, and running the lights for a few hours, it was apparent that we had a faulty set, as about a quarter of the lamps had blown in no time. After a while longer, about half of the lights were out (it was a 40 light set). The fuse lamp failed to operate, despite the operating tolerances of its spec being far exceeded. I didnt leave them plugged in any longer, since the lamps were getting very hot.
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Nowadays, the sets are made very cheaply, even filament sets dont have replaceable lamps, and they dont (in my experience) have secondary links, so once a lamp blows, its circuit is dead for good (no good when you are trying to create a pattern, or its those "snow" effect lights).
LED lamps are the way forward. Though B&Q seem to have excessive stock of old filament lights from last year, reduced to clear.