When importing attributes that are marked to allow only one per record, an exception will occur if a constituent record already has the attribute present:
Validation Error: The attribute "[Attribute Category]" has been configured to only allow one per record. You cannot add this attribute more than once.

If the attribute does not exist on the constituent record, it will import successfully.

To avoid the exception when importing, go to Configuration, Attributes in The Raiser's Edge to unmark the option "Allow Only One Per Record."  Please note unmarking this option will allow multiple attributes on the constituent record.