When attempting to resolve mail server errors/issues, being able to test the connection to a remote SMTP server is a critical step.
Testing a remote SMTP / mail server connection should be done from both a local machine (computer) and the server experiencing the issue.
From your terminal (shell) window perform the following:
dig MX hotmail.com +short 5 mx2.hotmail.com. 5 mx3.hotmail.com. 5 mx4.hotmail.com. 5 mx1.hotmail.com.
telnet mx1.hotmail.com 25
Trying 18.104.22.168... Connected to mx1.hotmail.com. Escape character is '^]'. 220 BAY004-MC4F30.hotmail.com Sending unsolicited commercial or bulk e-mail to Microsoft's computer network is prohibited. Other restrictions are found at http://privacy.microsoft.com/en-us/anti-spam.mspx. Sun, 1 Feb 2015 08:27:01 -0800
Trying 22.214.171.124... telnet: Unable to connect to remote host: Connection refused
It's not my preferred method, however you can always try external services for SMTP server diagnostics.
Vist MXToolBox's Email Server Test and provide the remote mail server (eg. mx1.hotmail.com) for a diagnostic report.