Jeep Repair Problems? Ask a Mechanic for Answers ASAP
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I'm not sure how the alternator was tested, but the only time it can be reliably tested is in the vehicle with the engine running and the battery light is on. If it was tested off the vehicle that's never a usable test for an intermittent problem, and if it was tested when the battery light was off then obviously it would have been charging fine.
The problem is either the alternator or it's wiring, it's quite easy to test the system at the time it's acting up. Nearly all charging problems such as yours are simply caused by a failed alternator. If you have a test light there is a terribly simple test you can do at the time it's not charging. Unplug the two wire field connector on the back of the alternator and insert a paperclip into one of the terminals. Attach the alligator clip to the paperclip, and then touch the second terminal in the connector with the probe end of the test light. If the test light lights up but the alternator isn't charging then it's bad and needs to be replaced.
Absolutely, they normally charge intermittently before failing altogether, just as yours is doing. 99% of the time your symptoms are caused by a failing alternator.