The full error:
Error sending using dbMail:
The mail could not be sent to the recipients because of the mail server failure. (Sending Mail using Account ........). Exception Message: Cannot send mails to mail server. (Failure sending mail.).)
These errors are probably happening as Microsoft have tightened up on their eMail sending rules and limits and you have probably fallen foul of these new rules.
The best solution would be to speak to your IT Provider to either set up a relay server through which to send the eMails or to look at using a different SMTP eMail sending service that has more relaxed rules.