A: 

If your running Windows XP the following should solve your problem.

Try setting your network passwords using Stored User Names and Passwords app.

http://support.microsoft.com/kb/306992

A: 

This may happen if you have manually copied source files to the 12 hive folders. open the file in Windows Explorer, and verify that it has the same (NTLM) security settings as the other files in the same folder.

Øyvind Skaar
A: 

In the specific case that you are getting prompted 2 (two) times, it could be that first time it is with the IP address and second time it is with the server name.

+1  A: 

Using Fiddler which is a web inspection tool, you can actually determine which domains the rejected requests are attempting to access. As Kinjal mentioned, it is possible that some of the images, css, or javascript requests are attempting to access through the ip, and not through the domain name.

MacGyver's option is possible, although that merely masks the problem, in that the multiple domains (or ways of accessing the same domain) are still being requested, but you are hiding it by caching the credentials that are to be used. This solution is also inadvisable if you are the one who is developing the solution, or administrating the farm, as you (usually) cannot possibly apply this hotfix to all the machines in your network.

The permissions possibility (as Øyvind points out) should also be verified, by checking which files are being rejected, and then verifying that the App Pool user has access to each of those locations.

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