Are the cables new? If not, there could be corrosion in between the connections under the insulation. I have not seen this happen personally, but have heard of it happeneing on really old cables.
Possibly a worn ignition switch too, or the contacts are a little corroded and when you go to start it there isn't enough material making contact on the connections, and when you turn the switch enough times it stops clicking and allows enough juice through to turn the starter.
As mentioned, a good starting point is to first check/replace the starter solenoid.
