Motor trouble question

Friend suspects a bad alternator in an '04 60 hp Merc four-stroke. If he starts the motor and then unhooks a cable at the battery, should the alternator keep the motor running?

The battery was tested by an auto mechanic and tested as good. After a short time of running, the motor will die and the battery will not turn it over. He switched to a TM battery and motor would run for a while and die. Battery would not turn motor over. All batteries were tested as good.

Any other ideas? Thanks.

Joe

if it has an ecm "computer" on it it will not keep running,when the voltage drops below probably 11volt it will not fire the computer to keep it running.
how old is the battery? i have them test good at autozone and others and still not be good.could be a voltage reg. problem.