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There is no reason why a person diagnosed with APD couldn't deteriorate psychologically or even develop an illness such as Depression (in fact APD often occurs co morbidly with Depression). People with APD have great difficulty with empathy but there is nothing in the diagnostic criteria that suggests that they couldn't 'break down' so to speak.
I hope this has been of some help, let me know if you have further questions or would like me to clarify anything that has been posted.
Good question by the way!