Indeed that might be possible. In fact there may be sub categories within the popular categories with different behaviors, e.g., a poet whose symbol is INTJ might be a different personality than an engineer with the same symbol. This would be hard to sort out because there are vastly fewer poets than engineers.

Intelligence may play a role too. Two people with the same symbols but widely different intelligence may behave differently to the point where they are identified as having different personalities. I'm speaking broadly about intelligence without necessarily believing in IQ testing.