I haven't had any problems logging into - or running scripts against - multiple databases on multiple servers.
That is, until I tried:
mysql> select state from state into outfile 'mikestatelist';
ERROR 1045 (28000): Access denied for user 'essnadmin'@'126.96.36.199/255.255.192.0' (using password: YES)
The query works fine without the 'into outfile' clause.