Hi Blake,
To add to Remus's information. Are you using the custom domain? If so, this issue is mostly caused by MX record.
To resolve this issue, you can do the following steps:
1. If you are using Microsoft Office 365 for professionals and small businesses, you can change name server records at a domain registrar.
2. If you are using Microsoft Office 365 for enterprises, you need to change the MX record to Office 365 at your domain name registrar. You can refer to this link below: onlinehelp.microsoft.com/.../jj655360.aspx
Note: If this domain exists elsewhere, you need to check and verify the domain is set to Shared. You can sign in Exchange Control Panel (ECP) > Mail Control > Domain and Protection > Check your domain.
In addition, to troubleshoot this issue, you also can refer to this link below:
community.office365.com/.../inbound-mail-flow.aspx
@ Remus thanks for your useful reply.
Best regards,
Jerry Cen