Again, not necessarily. Not all anxiety sufferers have the same symptoms. All chronic fatigue sufferers mostly do. So as to anxiety, you're overgeneralizing. The professional who can diagnose an anxiety problem is a psychologist; the professionals who can diagnose chronic fatigue syndrome is a physician.
I should add, chronic fatigue syndrome has a certain set of symptoms that leads to the diagnosis. But anxiety can manifest in a host of symptoms or very few. Having anxiety doesn't guarantee inability to sleep or fatigue or any particular symptom -- the symptoms are quite individual.
People who have chronic fatigue syndrome aren't necessarily anxious, at least not the chronic anxiety that is the mental disorder. If you're tired from anxiety, the anxiety would come first, then fatigue over a period of time. But often fatigue is caused by depression, which often goes hand in hand with anxiety.