Hi Paul,
To help resolve this issue, could you provide the following information for us to search?
1.Does the specific user can receive emails from other external users?
2.Please let the user to login to OWA(Outlook Web App) to see if he can receive emails from this external user?
3.Is there any other people who also encountered the same issue in your company?
4.Please contact the external sender to see if he receive an non-delivery report(NDR) to notice him the email has not been sent out successfully. If so, please post the whole error message here for us to troubleshoot.
Additionally, please let the user add the external users as a safe sender by referring to the steps below to see if this will help:
1. Log in to Office 365 and click Outlook from the top header.
2. Click Options in the upper right corner and click See All Options.
3. Under Mail > Options: Manage Myself, click Block or Allow.
4. Add the email addresses of the external user as a Safe Sender.
5. Click Save and check the results.
For more information about Safe Senders and Recipients, please refer to the link below:
http://help.outlook.com/en-us/140/ms.exch.ecp.junkemailtrustedsendersanddomains.aspx
Furthermore, if you use Office 365 for enterprise (Plan E), you can perform a message trace in FOPE by referring to the link below to identify problems with message routing.
How to perform a message trace in the Forefront Online Protection for Exchange (FOPE) Administration Center:
http://support.microsoft.com/kb/2690809
Thanks,
Linda Wang