After receiving the last two nixie driver IC's, and finally being able to run all four tubes, it turned out that my crude breadboarded HV PSU wasn't quite up to the task. The inductor on the PSU was getting very hot and the HV supply was sagging trying to keep the tubes lit.

It turns out that LV to HV switch mode PSU design is not that trivial if you want an efficient design, and hacking such a PSU together on a piece of breadboard is definately a no-go.

Instead of wasting more time on making a proper HV PSU, I ended buying a kit like this:

The kit arrived a few days later and with that assembled and having replaced the old PSU, all four tubes where lit without any problems or excessive heat.