If you see the following error in the event log:

Event ID: 12289
Source: VSS
Volume Shadow Copy Service error: Unexpected error RegOpenKeyExW(-2147483646,SYSTEM\CurrentControlSet\Services\VSS\Diag,...). hr = 0x80070005.
Operation:
Initializing Writer
Context:
Writer Class Id: {8d5f38cd-fb7a-49ca-ae1b-21d1f017d8f9}
Writer Name: OSearch VSS Writer
Writer Instance ID: {55883ff4-1cba-4f0f-9e44-2b4d398751ef}

Open the registry (Start > Run > "Regedit") and browse to Computer > HKEY_LOCAL_MACHINE > SYSTEM > CurrentControlSet > Services > VSS > Diag and right-click on "Diag" and choose permissions. We need to add the "Office SharePoint Server Search Service" account with Read Full permissions (you can confirm this account by looking at Start > Run > services.msc and check the "Log On" account for the "Office SharePoint Server Search Properties").

OSearch VSS Write is used by the Office SharePoint Server Search Service (as error above)
SPSearch VSS Writer is used by the Windows SharePoint Services Search Service (wasn't running on our environment)

Apply to all servers on the farm (probably only needed for the search servers).