Let's start with theory and practical (A3S trap in hand).
This infomation is a review of Electronis 101 -- BASIC RF Knowledge required to be known by amateur radio amateurs. IF you skipped it, now you know the shortcomings of memorizing answers.
The Cushcraft A3S is a Multi-band TRAP antenna.
The traps are an L-C component (L = Inductance; C=Capacitance).http://en.wikipedia.org/wiki/LC_circuit
I decided to take the trap off and inspect it again. It still looked fine and the resistance reading was less than 1 ohm. I decided to put it back up WITHOUT the aluminum casing, just to see what would happen. No change...but when I brought it back down, put it back in the aluminum shell and reinstalled it on the antenna, the SWR came down to around 1.2. So I am figuring that maybe I did not have a good enough contact with the tubing and the problem was not the trap after all. Not really sure.
The Inductance is 74 turns of 12 AWG wire (HV enamel insulation) on a 3/4 inch non-conductive (plastic) form. BUT, Where I the Capacitance? We will get to that in a minute.
Let's first comment on your diagnostic tool, a DVM or VOM set to DC continuity.
That is fine to tell you that there is no breaks in the 74 turns of 12 AWG, BUT that is NOT enough !!
RADIO FREQUENCY (RF) follows AC THEORY, NOT DC THEORY !!
A GRID DIP METER (GDO) is the diagnostic tool of choice for L-C TRAPS.
NOW, back to your trap's Capacitance. The outer aluminum tube (shell) of the Trap is the outer shell of a COAXIAL CAPACITOR !! That metal casing is an active component of the trap.
SO, a poor or loose mechanical or electrical connections between that outer tube and it's connection to the inner aluminum element tube -- means you have no capacitance OR a value different than desired (bad capacitor) -- the L-C trap does not work!
LAST, in the area of the trap (Inductance) where you had ARCING (discoloration) that your neighbor saw!!
Carbon tracks from those HV arcs produced NEW conductive paths (NOT desirable).
In this case you should clean, and apply a HIGH VOLTAGE Insulating Varnish (High Corona Dope) to prevent future arcs in this area.http://shop.willyselectronics.com/browse.cfm/mg-super-corona-dope-high-voltage-insulating-varnish-2-oz-/4,7361.html
Currently, there is no insulation inside that trap anymore but I am getting SWR around 1.4 at 14.010 up to 2.5 on 14.250. Not too bad, really. But is the insulation required? What does it do?
Insulation can be reapplied, new adhesive heat shink products are very good for this application.
Readily available from McMaster-Carr, Grainger, and major electrical/Electronis suppliers.