Have you swapped out any images or applied css files ?

This happens a lot when images are swapped or css files appled and users
do not have ntfs read access to those files on the web front end hard
drives

I haven't customized anything yet.

One of my coworkers said he got the login  box when he was uploading
files to a document library in his My Site.

the thing about Kerberos is with the clock times - i would check
the time on each machine (including the users) to see that they have the
same time / time zone - ad should sort this for you but you never know

What I'm not getting is why it asks you to log on for the mysites - I
mean, why it asks you to log in to upload to mysite but not to log on
for the my site itself - you get this uploading files where the site
isn't in your local intranet zone so you could check that you have added
it there ... and also that its added in the way that users  are
accessing it (i.e. are they use a fqdn or just the server name ?)

Other than that, I'm a bit stumped - are all users being asked to log on
when they upload ? i.e. is it when they upload that they're being asked
or is it at other times ? is it uploading to my sites or uploading to
other places ? What I'm after is "IS there anything common among the
users being asked to log on once again" ?

Verify both the Server(s) and the clients see the KDC.

I asked my Network Engineer about the clock times, and he said they
were fine.

The problem appears  to be inconsistent. One person gets the login
box when she tries to open certain documents, which she should have
rights to as a member. I get the login  box sometimes when I refresh
my page after making changes to a list that has a webpart view.
Others may have the site open and minimized, then they try to
navigate later and receive the login box.

I've asked my NetEng. to verify the KDC as well, per Ben's advice.
He's looking into it still.

As a team, we are new to AD and Kerberos. We implemented AD last
fall and are finding that a lot of our systems have to be adapted to
this new environment. We don't have any real experts here, which
makes it more difficult to troubleshoot.

So, another question ..... when you get the logon box  have you "left"
the browser open for some time and go back to it or do you get the logon
prompt while you are actively using the session (i.e. you click
something, fine, click something, fine, click something, fine, click
something, logon box) ?

In fact .... i see you answered that below !

Do you use a proxy to access the internet and are you by-passing it for
internal sites ?

The people who get promoted to log back in after an idle period is quite
normal and the amount of time will depend on your idle timeout either in
IIS or your proxy and I think there is also a moss setting.

The lady who gets the prompts when she access certain documents all on
one web application (i.e. can she get in to one web application and not
another) or are you just running all the sites under one web application
?

We're not using a proxy, but we are using one web application. Our
group is using a Team Site Collection that currently has about five
subsites. This login  box most commonly appears  when people are
working with documents, whether uploading or viewing. Just this
morning I opened a document in one of our document libraries and
received the login box. When I typed in my credentials (trying both
fully qualified and not), it did not resolve. Interestingly, I
received an error I haven't seen before. The VB debugger launched
and displayed an "ambiguous credentials" dialog. After a few
attempts to open the doc, I restarted my pc. I was then able to open
the doc successfully.

I've seen the timeout settings in Moss (currently 30 minutes). But,
it's disturbing that when the login box  appears, it doesn't resolve
the login. I have to close and relaunch my browser to fix it.

In addition to my last post below...I just experienced the login  box
after an idle period. I had my Sharepoint team site open/minimized.
I went back into edit it and the login box  appeared. Again, it would
not resolve my login after many attempts, which ultimately bring me
to an authorization failure page ("you are not authorized to view
this page"). However, if I close and reopen my browser, it's back to
normal.

Does anyone know where I can go to recruit a Sharepoint administrator in
the UK on a contract basis.

Are you prefixing the login  with the domain name?

As in: MYCORP\janedoe

Once you successfully login, it shouldn't display the login box  anymore, unless
you have Enable Integrated Windows Authentication unchecked in the advanced
internet options settings (assuming IE 7).

I wasn't expecting to ever get a login  box. But when the login
appears, I've tried the login with name only and with the domain to
no avail. We're using IE 6.0.