I solved your problem. Let me guess, you didn't assign a Z Coordinate System for the Feature Dataset that contained your road network layer? Was having the same problem with the Dissolve Network tool falling over at 'Dissolving feature chains', and I also got the corresponding '030083' error. Tried changing the Dissolve Network tool's environmental variables so that 'M Values' and 'Z Values' were set to 'Disabled', but that didn't work. I then tried adding a proxy Z Value column to my roads layer with all features assigned a value of 0, and setting this as the Z attribute. That just threw up an entirely new error message.
Finally, I tried assigning a Z Coordinate System for the Feature Dataset that contained my road network layer and now it is working. I did not change any of the environmental variables. Turns out the GP tool was looking further up the chain for the Z geometry.
You are welcome