The "Import Multi-value Custom Data" fails when importing data into a multi-value custom field set that applies to only a Household contact type or Household sub-types. When the field set was change to "All Contacts" the import works. I didn't test it with multi-value custom field sets restricted to "Individual" or "Organization" contact types. It may fail with these also.
The Import Errors file show "Invalid value for field(s) : field ID; ; field ID; ; field ID; field ID" as the reason for the error when four fields are imported.
We use CiviCRM for WordPress.
To duplicate this setup a simple multi-value custom field set restricted to Households. Then try to import data into it.