Administrative Messages won't Send

@Dougster I think that all system emails are sent by the function sendMailPhpMailer in file admin/lib.php around line 427

If you are sending through an smtp server then you can enable debug, which might show what is not working

add these lines

$mail->SMTPDebug = 3;
$mail->Debugoutput = 'html';

immediately after this line which is line 454

$mail = new PHPlistMailer('systemmessage', $destinationemail, false);

Then generate a system email, such as by adding a new subscriber using a subscribe page, while signed-in as an admin. There should then be some output showing the exchange with the smtp server.