If you can start the engine by placing the jumper cable between the battery terminal and starter solenoid terminal where the battery cable attaches (bypassing the cars battery cable), check for problems with the cable.
Such as poor or corroded connections between the battery cable and its end fitting terminals that attach to the battery and starter solenoid or a bad connection between the battery cable terminal and the battery terminal.
The large cable on the starter solenoid goes directly to the battery, nothing is in-between.
Use a volt meter to test the cable as described in the line below:
How to Check Resistance in Battery Cables | eHow.com
Voltage Drop Testing
Connect the meter leads between the terminals ends of the cable (they are still connected). This will measure the voltage drop across the cable. Too high of drop will result in lack of available current at the starter solenoid. Jumping the battery with another battery provides extra available current.