Probably more than one. When one of the "If one bulb blows, the rest stay on!" strings has one blow, yes, the rest of the string stays on, but it's the first domino that has fallen.

Let's say there are 50 lights in that string. That string is designed to hold X amount of current, which is divided evenly among 50 bulbs. When one blows, there's still the same amount of current going through the circuit, but it's now running through 49 filaments instead of 50. There's a small incremental increase going to each bulb. This puts higher strain on the weaker filaments in the string, and pretty soon another blows. Now you're putting 50 bulbs worth of current through 48. And then 47, 46, 45 until the entire string blows.

I've actually gone through about 1/3 of an old tree that we had replacing entire branches worth of bulbs. Take a look around that entire section that is out. If you notice more than one bulb with the telltale black film on the inside, you're probably looking at replacing every bulb in that section.