>...you cannot get the bends if you're not breathing.
Whether you are breathing is not fundamentally the issue - what matters is what gases you have in your lungs, under what pressure, and for how long. If you have nitrogen in your lungs, it will diffuse into your bloodstream, and thence to tissues, until the partial pressures are equalized. This is not normally a problem for free divers - even most deep free divers - because the combination of time and pressure are not great enough, but it seems that Nitsch might have crossed a line here.
I tried doing a back-of-envelope calculation to see if this seemed plausible. No-decompression limits in dive tables for air breathing bottom out at 5 minutes at about 60 m, and while the depth cutoff is presumably set by nitrogen narcosis and/or oxygen toxicity, it is clear that nitrogen absorption is quickly reaching the bends-risk level (for comparison, this dive reached a depth of 253m and lasted about 4 1/2 minutes.)
There are several factors to be taken into consideration. Firstly, Nitsch's lungs would have been considerably collapsed at depth, and so presumably less-effective at gas transfer. secondly, dive tables are conservative. On the other hand, though, they are based on the tissues in which the diffusion is slowest (if it was effectively instantaneous, there would be no risk of the bends on ascent) and assume the diver is ascending slowly even when they are not taking decompression stops, so maybe in Nitsch's case, the problem arose in different, faster-diffusing tissue (neural tissue?) than that on which the tables are based.
Given all this, I'm inclined to believe Nitsch's own claim that he had a case of decompression sickness from this dive.