Adrenal Fatigue is the result of Chronic Stress, not the other way around.
Actually it goes both ways. Stress weakens the adrenals as I pointed out. But weak adrenals reduce the levels of the body's anti-stress hormone, which in turn makes people stress easier further weakening the adrenals. This is why one of the symptoms of adrenal fatigue is "flying of the handle" even with slightest provocation. When the adrenals are healthy they release anti-stress hormone when we are exposed to stress to keep us on an even keel. When the adrenals are too weak though they do not respond with sufficient levels of anti-stress hormone and people over react to the stressful situation. This stress further reduces the adrenal's ability to help people cope with stress.
In addition, adrenal dysfunction has more causes than simply stress. Over use of steroids or stimulants for example are a few other causes of adrenal dysfunction.