Let's take a step forward and see if I understand this damned series lamp: when I don't have any short circuits on the board under test, the lamp remains off, meaning the filament is cold and its resistance is close to zero. In this case, if the board requires 1A, the current flows easily through the lamp.
Let's say I have a short circuit on the board, requiring 1.5A: if I have a 60W lamp, it turns on and lets a maximum of 1.3A through, limiting the current to the board (though we'll still have an excess of 0.3A on the board). If I were to use a 100W lamp, it turns on and lets a maximum of 1A through (the board's nominal voltage, so better).
Have I taken a step forward, or two steps back? 
Thanks
The first part is kind of correct. The resistance of the lamp is low enough but it depends on the wattage of the bulb. I just measured two 40W bulbs and they showed about 95 Ohm, so the startup current is ~2.5A. The nominal current is only 0.1666A which means a resistance of ~1440 Ohm. (This based on 240V)
As long as the target board draws significantly less than that 2.5A the lamp resistance will hardly change and the voltage drop over the lamp will be low enough. But it very much depends on the device under test. In case of a larger inrush current the device might never get to a normal point of operation, because the lamp will be limiting both the current and the voltage.
The simple theory is two series resistors that as such make up a voltage divider. Both resistors are in play as to what the current is going to be and who gets the biggest part of the voltage supplied. With one or both of the resistors having a large dependency on the temperature the result will vary over this temperature change.
For the second part, if there is something wrong with the board and it takes a higher current than normal, based on your premise 1.5A, the lamp will initially take that 1.5A and start to warm up and increase its resistance. This will cause the current to drop. The device might then not act as a short anymore and let the current drop a lot more, cooling down the lamp. This leads to a voltage rise on the device and it might act up again, etc.
It is a dynamic system at play here. Based on the lamps wattage it is possible to calculate the nominal current of the lamp. In case of a real short in the device under test, it is this current that will be flowing once the lamp is fully on. The benefit of the lamp in series is that in case of an actual short, the initial current is limited to the startup current of the lamp and not a high enough current that can evaporate circuit board traces.
This is a problem with fuses. Even though a fuse for instance is rated for 1A, it does not limit the current to 1A. It takes time to blow a fuse, which depends on the type of fuse. Slow or fast are two types, and putting a fast fuse in a device with a high inrush current will blow it every time the device is powered on. In the case of a fuse and also the lamp, the max current depends on the total resistance of the circuit from the source to the device. Wiring has resistance to.
Hope this clears things up a bit for you.