Hi taz92,
TLS is required for both the Send connector and Receive connector configuration in Hybrid environment. Please check your transport settings by the Configure Mail Flow for an Exchange 2010 Hybrid Deployment or Exchange Server Deployment Assistant, and make sure the TLS has been enabled correctly and port 587 is opened in your firewall.
In addition, Hybrid deployment needs a third party public certificate for the on-premises Hybrid server, please use the Microsoft Remote Connectivity Analyzer to check the RPC over Http of the Hybrid server, and ensure there is no any certificate error.
If the issue persists, I would like to confirm the following things for further investigation.
1. Has the tenant upgraded or during the service upgrade?
http://office.microsoft.com/en-in/help/am-i-using-office-365-after-the-service-upgrade-HA103982331.aspx
2. What’s the version of Hybrid server? Are there any other Exchange server in your organization?
3. How long does the issue last?
Thanks,
Johnny Zhang