I would suspect that the problem lies in the power source. In order to vaporise the nichrome wires the voltage needs to rise to something significantly greater than intended. For a reasonable range of voltages the nichrome wire will provide a negative feedback effect, rising in resistance and lowering the overall amount of power being dissipated. Clearly the voltage delivered is more than just a small jump over the amount desired for operation.
With a perfect power supply everything should work fine. A constant voltage source will deliver whatever current each leg of the device draws, and there should be no failures. In reality all power supplies have some internal resistance. Regulated supplies have effectively a very small source resistance, mostly due to the resistance of the leads.
Laboratory power supplies are usually notionally constant (if variable) voltage, and many have settable current limiting. Such a device should cause no problems.
Other power supply possibilities include a rheostat, or Variac. (A potential divider isn’t going to be useful.)
A rheostat is just a series resistor, the more parallel nichrome wires there are and the greater the voltage drop across the rheostat will be, requiring it to be adjusted for each setup. But add more wires, and for the same setting, there will be lower voltage on the output terminal of the rheostat, not more.
A Variac has a mix of reasons for variation of output voltage, but again, the lower the resistance of the nichrome wire setup, the lower the output voltage will be for a given setting due to internal losses expressing themselves as a series resistance.
One out-there idea is if a Variac was used, and the input current limit was reached, if the Variac had a thermal cut-out on the input, it would open circuit the Variac suddenly, which might lead to a significant inductive spike being delivered to the apparatus. However I’m far from convinced that there would be enough energy in the system to vaporise the wires. Such a scenario is a bit of a stretch, but not impossible. I would also be really worried that anyone would use a Variac in such an experiment. They are not designed for such use, and are intrinsically dangerous devices, being quite capable of electrocuting the unwary.
A cascading failure brought about from a poorly behaving rheostat, one with jumpy behaviour may well be the problem. Set at one resistance, good for a single wire, but unable to manage a smooth transition to the lower resistance needed for a set of parallel wires. So you turn it on, then up, and then - bang - it jumps down in resistance to something fatal and the whole thing blows.