I encountered an issue recently where, in an Exchange 2010 / Office 365 Hybrid environment, Office 365 users were unable to see calendar and Free/Busy information from On-Premises mailboxes.
I ran through a variety of diagnostic steps, including the “Hybrid Environment Free/busy Troubleshooter” https://support.microsoft.com/common/survey.aspx?scid=sw%3ben%3b3526&showpage=1 and the Free/Busy test on the Office 365 of the Remote Connectivity Analyzer https://testconnectivity.microsoft.com/
Neither tool registered an error or indicated a configuration problem.
Eventually, I took a look at the state of the Federation Trust using Test-FederationTrust from the On-Premises Exchange server which returned:
Shane Skriletz , PEI