I had this problem with one of my office 365 costumers when i tried to send emails from on premises mailbox to the cloud and got this NDR:
Your message did not reach some or all of the intended recipients.
Subject: test from on premises 4
Sent: 2/27/2013 7:10 PM
The following recipient(s) could not be reached:
i386 user1 on 2/27/2013 7:10 PM
A configuration error in the e-mail system caused the message to bounce between two servers or to be forwarded between two recipients. Contact your administrator.
<exh2003.domain.com #5.4.6>
Solution:
First I created on the Exchange 2003 server an SMTP connector that will route any mail destined for *.mail.onmicrosoft.com to the hybrid Exchange 2010 server
Then I changed the tenantID.mail.onmicrosoft.com SMTP address space to Internal Relay instead of Authoritative Exchange 2003 will send the message through the Exchange 2010 server using the TLS secured connection.
Exchange 2003 connector properties :
Exchange 2010 – Accepted domain
Good luck!
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