There aren't any logs that you'd be able to see in the software, however you should be able to find any error messages on your mail server. Have you tried placing a test order to see what happens? The mailing test manager uses email addresses that are specified in the appconfig parameters so there's a chance that these are all internal (eg. on the same domain) as the mail server so there's no problem sending but the emails can't send to email addresses outside. Maybe check your mail server password to make sure it didn't expire and that you don't have to change it, and then finally check your mail server logs to see if anything points to the cause of the failure.
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