Like most things, there is a predefined default, and at some point in time, there will be a need to deviate from this. Office 365 Admins should have an understanding of those and setup procedures to increase the size for individuals or the organization when the need arises

To increase the Office 365 Mailbox size we must use Windows PowerShell as the Web UI lacks the ability to make such changes at this time.

Step 1: Connect to Office 365 – Exchange Online:
An Office 365 Exchange Online Admin Account is Required

$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
Import-PSSession $Session

Step 2: Get information about the mailbox you are looking to increase size on. I like to take note of the results before and then after I introduce changes

Get-Mailbox jermsmit | Out-GridView
Get-Mailbox jermsmit | Select *quota

Step 3: Issue the command to increase the storage:
Example: Set-Mailbox -ProhibitSendQuota -ProhibitSendReceiveQuota -IssueWarningQuota

Set-Mailbox jermsmit -ProhibitSendQuota 69.5GB -ProhibitSendReceiveQuota 70GB -IssueWarningQuota 69GB

Step 4: Review your changes, for this we goto step Step 2

For more information on Office 365 Mailbox Limits, please visit the following: https://docs.microsoft.com/en-us/office365/servicedescriptions/exchange-online-service-description/exchange-online-limits#mailbox-storage-limits

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