Corona Discharge Current, Atmospheric pressure discharge!
If there is no effect from the setup itself, this sounds like an effect of the ionized air to me.

In a corona discharge, the electric field is high enough to ionize the air, making it a conductor and creating light.

If the voltage is switched off, there are still all the ionized atoms around the wire, and it takes time for them to recombine with electrons. This time typically is in the order of milliseconds. This explains why you can see an effect when both pulses are generated within nanoseconds, and not, when they are generated within milliseconds.

I'm not sure why the current of the second pulse is lower, one could also imagine a higher current, since there are already ions around. It is possible that the ions already act as conductor before the voltage is high enough for a corona discharge, which means the effective diameter of the conductor (the wire) increases. This lowers the field strength and so the rate of ionization and the current flow.

Think about: 100A is really lots of current, and I can't imagine a corona discharge with a constant current of 100A in a smaller setup. This rather seems to be an inrush current.

Is it possible to generate a constant 16kV with a short leading edge? I guess you'll see a high current peak falling to a much lower constant value. 

But as said, also check your equipment and make sure it doesn't affect your measurement. (For example, use a much thicker central wire to prevent corona discharge and compare)