John,

Where is the backup file stored?  Apparently box # 1 has access, but box #
2 does not.  Is the backup file on box # 1 or somewhere on a network?  If on
the network, how did box # 1 get access to that location?

I understand that you logged in through BUILTIN\Administrators, but under
what account is SQL Server running?  Is it Local System?  NT
Authority\Network Service? Et cetera? If so, that is a login local to the
box it runs on and does not have rights on some other box.  Ideally (I don't
know your setup of course) the SQL Servers should be run as a domain account
that can be granted the needed rights to the network, other boxes, etc.

Here is the Books Online consideration of your service account choices and
the impact of each:
http://msdn.microsoft.com/en-us/library/ms143170(SQL.90).aspx

RLF

John
In addition to Russell's comment
Can you issue a backup  from one box to another  like

BACKUP DATABASE dbname TO DISK =
   N'\\servername\shared_folder\dbname.BAK'

Note: make sure that box 1 'sees' the box 2 via shared folder