Sorry, I said App Notes, but it was actually the datasheet (
http://cds.linear.com/docs/Datasheet/1073fa.pdf) that helped me out. The IceNINE circuit is a copy of the App Notes circuit. The datasheet circuit is functionally similar, but with different values for the voltage supply caps. The first specify 0.47uF caps, the datasheet actually specifies 0.1uF caps. It made a real difference changing the one that is connected to the inductor to the lower value cap (the other two didn't seem to make as much difference). With the wrong caps, the voltage into the transistor was far too low -- something like 20v, IIRC. With the correct value caps, the voltage rises nicely above 90v, which is critical to get the sought-after pulse. At least, this is what happened in my case. YMMV.
One thing the app notes pointed out was the importance of keeping the leads as short as possible. I had much less ringing when I clipped the leads off so they were just long enough to reach the holes in the breadboard. Now that I have the circuit working, I'll try soldering everything direct to the BNC connector, as the app notes suggest.
If I understand the principle correctly and assuming I am actually generating a pulse with a 350ps rise time, then the actual bandwidth of my particular scope with the supplied probes should be (0.34/2.12)*1,000 = ~160MHz. Not bad for a 50MHz price!