I tried executing the following Bulk Insert command, having UNC paths, from SQL Server Management Studio running on my laptop and connected to the appropriate database.  I connected using Windows Authentication.

BULK INSERT dbo.Load_itemDetail
FROM '\\server1\Outbound\orditem.txt'
WITH (FORMATFILE = '\\server1\Outbound\FormatFiles\ORDITEMNEW.FMT')

I then received the following error message:

Msg 4860, Level 16, State 1, Line 2 Cannot bulk load. The file "\\server1\Outbound\orditem.txt" does not exist.

When I had our administrator run the bulk insert command from within SQL Server Management Studio running on the same machine as the SQL Server instance, the command worked.

Fix: When running SQL Server Management Studio from a machine other than the machine where the SQL Server instance resides, you need to connect to the server where the database resides using SQL Server authentication.  The SQL Server login account that you use to connect must also have the "bulkadmin" server role assigned to it.  After logging in with SQL Server authentication, I was able to execute the Bulk Insert command without error.