I have seen this many times and sometimes the fix does not exactly seem to make sense to most folks like a loose connection.
try this, if you can swap batteries with a known good one—a battery can cause this
look and feel along the cable for bumps—the cable will swell if corroded internally
even if you think they are good clean the posts and the terminals and apply a little smear of vaseline on them or a special terminal grease
dont forget the connection to the starter for a similar treatment
check the ground strap between the engine and the frame
check the wires on the starter solenoid (small thing with small wires attached to the starter)
finally I would change the starter and solenoid although the fault is probably just the solenoid
a jump gives you 14 volts whereas a battery only gives you 12 volts or less if the battery is low
beyond that google starting system trouble shooting and voltage drops checks and ohms checks for more info on tracking down the prob

{ 4 comments… read them below or add one }
Are you jumping directly to the battery posts, positive to positive and negative to negative? Or are you jumping to a hot terminal between the battery and the starter, with a ground somewhere on the metal?
If you are, you are not jumping the battery but starting your car off of someone else's. One thing that happens is the plates in your battery corrode and will not allow you to jump them.
Hook your cables up to positive and negative directly on the battery. If it will not start, your battery is bad due to plate corrosion.
A battery can have proper voltage and still be no good due to plate corrosion. When you lose a plate you don't have the power (amps) that you need to turn your starter.
Sounds like a bad ground to me. It's grounding through the jumper cables and the other car.
Clean your battery terminals. DO NOT grease them. Jumper cables by-pass the terminal connection from cable to battery.
buy a new battery and then retest everything…
Sometimes, when a weak battery sits, like overnight, with all of those little electrical devises still powered up ( engine/body computers, radio ,memory ) it looses enough charge to be able to start the vehicle the next morning, especially on cold nights…… When you take it in to test, they have to make sure it's fully charged or the test the try to do isn't accurate…
You can simply check a battery under charge, watch the voltage and see how high it goes, too high and it's bad… Anything over 15.5 volts is when we start to consider it bad…
Bad connections at the battery can also cause a condition like this, check those first always
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