Please try using this CVS snapshot:

  http://snaps.php.net/php5.2-latest.tar.gz
 
For Windows (zip):
 
  http://snaps.php.net/win32/php5.2-win32-latest.zip

For Windows (installer):

  http://snaps.php.net/win32/php5.2-win32-installer-latest.msi

Upgrading to snap did not fix.


[root@devel1 src]# /usr/local/php/bin/php -v
PHP 5.2.7-dev (cli) (built: Aug 28 2008 20:54:37) 
Copyright (c) 1997-2008 The PHP Group
Zend Engine v2.2.0, Copyright (c) 1998-2008 Zend Technologies
[root@devel1 src]# /usr/local/php/bin/php -q test.php

Warning: mysqli::mysqli(): (HY000/2005): Unknown MySQL server host '127.0.0.1:3307' (3) in /srv/php/projects/test/src/test.php on line 4

Warning: main(): Couldn't fetch mysqli in /srv/php/projects/test/src/test.php on line 7

Warning: main(): Couldn't fetch mysqli in /srv/php/projects/test/src/test.php on line 12
Host information: 

Warning: mysqli::close(): Couldn't fetch mysqli in /srv/php/projects/test/src/test.php on line 15
[root@devel1 src]#

I can reproduce that, assigning to mysql folks.

Get the same problem but make investigation:

$mysqli = new mysqli("localhost:3307", "my_user", "my_password",
"world");
-> got error

$mysqli = new mysqli("127.0.0.1:3307", "my_user", "my_password",
"world");
-> got error

BUT!:
$mysqli = new mysqli("127.0.0.1:3306", "my_user", "my_password",
"world");
-> change server port to default (3306).

$mysqli = new mysqli("localhost", "my_user", "my_password",
"world");
-> all works correct. 
Seems that variable not parsed for $HOST_NAME and $PORT_NUMBER and function try to connect using incorrect (not parsed) string.

Not quite sure that mudroeb is right.  
According to manual the right call format must be following:
mysqli mysqli_connect ( [string host [, string username [, string passwd [, string dbname [, int port [, string socket]]]]]] )

However  
conn=new mysqli('localhost','user','password','my_db',3306);
works as expected.

I have duplicated this issue.

According to the manual mysqli::__construct should always return a object, but the function is returning a false value on error.

I have tried three different possible error conditions and have the following results:

   Bad user/password combo: mysqli_connect_error() returns "Access denied for user 'xxx'@'xxx' (using password: YES)", return value is false.

   User does not have access to the db selected: mysqli_connect_error()returns "Access denied for user 'xxx'@'xxx' to database 'xxx'", return value is false.

   Bad port number: mysqli_connect_error() returns "Can't connect to MySQL server on 'localhost' (10061)", return value is false.

I have not checked any other cases such as bad socked name, but in each of the above conditions I received a false value not an object.

test code:
<?php
    $dbObj = new mysqli( "localhost", "user", "pass", "db", 3306);

     echo "dbObj=";
     var_dump($dbObj);
?>

error result:
dbObj=bool(false)

sucess result:
dbObj==object(mysqli)#1 (0) {
}

Forget my previous comment.  I found some code between the new mysqli statement in the example I was using which was setting the $dbObj to false if mysql_connect_error was returning a value.

Was able to reproduce the error in the same method on php 5.2.8, both with purposeful incorrect passwords and nonexistent host/port combos.

the object is created, connect_error returns NULL on var_dump, error log dumps:
"Dec 25 15:12:18 dev01 httpd: PHP Warning:  dbo::__construct() [<a href='dbo.--construct'>dbo.--construct</a>]: Couldn't fetch mysqli in /nfs/beta/www/classes/dbo-new.inc.php on line 88"

Getting the same error with the following version:
PHP 5.2.4-2ubuntu5.4 with Suhosin-Patch 0.9.6.2

I was attempting to suppress errors so I could control output/formatting etc. Thought i'd just re-iterate this for version related purposes.

<?php
//a forced username/password mismatch
$dbo = @new mysqli('localhost', 'r00t', 'pass', 'my_db');

var_dump($dbo->connect_error);
var_dump(mysqli_connect_error());
?>

output:

Warning: main(): Couldn't fetch mysqli in /home/default/php/mysqli.php on line 4
NULL
string(63) "Access denied for user 'r00t'@'localhost' (using password: YES)"

This bug has been fixed in CVS.

Snapshots of the sources are packaged every three hours; this change
will be in the next snapshot. You can grab the snapshot at
http://snaps.php.net/.
 
Thank you for the report, and for helping us make PHP better.

Fixed in 5.2 CVS, too, now.

Added information to the documentation.