The boys are 100% spot on Brian. I’m a tad unclear about your one line Stu, but all of the lights in the truck are fed thru the thermal cut out including the dash lights and dimmer. Unless they are an accessory that is connected to the battery terminal or power buss at the back of the switch.
Brian, this is a common problem with the original GM light switches of the first half of the 50’s. There is a bi-metal strip that the contacts are mounted on. When the circuit draws more than around 40 amps it should heat up enough to open the circuit.
A lot of back yard mechanics mistakenly bend the mounting tab for the thermal strip thinking that if a little tension is good then a bunch more is better. This only throws out the calibration and allows a higher current to pass, thus over loading the entire circuit, your new wiring included. This is NOT a good solution. Also the phenolic contact board is made with a paper substrate and with years of exposure to moisture it swells and warps. Most of them that I have found are warped this way.
There is a high resistance somewhere in your circuit possibly at one of the connections. If you simply changed the wiring loom without thoroughly cleaning the oxidation from all of the mating metal points then there may still be a high resistance load at one of the new connections.
The other place to consider could be inside the light switch it self. The old grease migrates onto the internal contacts and dries out preventing them from making full contact. The copper contacts oxidize which again introduces a high resistance into the circuit and sometimes they become intermittent because of these problems.
I have been working diligently over the winter to develop parts and tooling to for new internal parts with the intention of rebuilding the original light switches. This would most likely only be of interest to those that are interested in keeping everything absolutely factory original. So far there really doesn’t seem to be much interest in this alternative since I have only been able to find a handful of cores to work with that have not been tampered with. This rebuild is very labor intensive and although it is possible, if you are very careful, to open the factory sealed head light section up, you will most likely find that the internal parts are not only dirty from dried grease but that the contacts and board have a groove worn in them or are burned or oxidized. And as mentioned above, that the contact board is distorted also and needs to be replaced.
The vendors sell new after market switches for around $20 and since it is behind the dash and out of sight no one will ever know if it is the original switch or not, so you can see why not many people are willing to go through the effort of restoring a light switch.
Well good luck with your problem, and if you do end up choosing to replace the switch, keep me in mind for the old cadaver.
Denny Graham
Sandwich, IL
If you care to see some pics of the switches internals here are a few:
http://community.webshots.com/album/555161678EhIoCb?start=288 
