i have had a problem recently with the starting system on my 68. sometimes it starts and sometimes i turn the key with dead air (no clicks or other noises) the batterys fully charged and the starter is new. i have checked the wires to and from the battery and starter and cant find anything out of place. is there a simple test i can do from each of the starting components in lieu of having to swap parts in and out?
Check the connector at your neutral safety switch. If you work under the dash it is easy to knock loose - mine is a two wire plug near the fire wall above the steering column - blue / blue-white wires I think. Ask me how I know . . .
The neutral safety switch for a 1968 console shifted automatic car will be in the console attached to the shift mechanism.
It may need an adjustment. Try moving the shifter around while the key is in the start position. Be carfull. If the car starts in gear it will take off on you.
If you have a manual transmission car try running a test light to the starter. Attach the light to the start trigger terminal (large purple wire). This way you can tell if the starter is getting the start/trigger signal.