in the connection file, the last line will use mysql_pconnect to make the connection to the database. for example, a connection named localhost will have the following code at line 9:
$localhost = mysql_pconnect($hostname_localhost, $username_localhost, $password_localhost) or trigger_error(mysql_error(),E_USER_ERROR);
some hosts don't like the use of persistent connections. it sounds like this may be one of them. you can change mysql_pconnect to mysql_connect (in other words, drop the "p") to fix it:
$localhost = mysql_connect($hostname_localhost, $username_localhost, $password_localhost) or trigger_error(mysql_error(),E_USER_ERROR);