If it's only a couple of users, I'd check that their LDSD account has a corresponding network account and that it is the right account. I'd also check that the people are on the right domain and are not coming in from an external location. If you turn on authentication diagnostics in the configuration center for the framework being used, this sometimes throws light on the matter too.
Or it could be that the browser is not passing the login details.
Have you tried those users on another computer where someone else has been working do they manage to login ?
Sometimes you can get the prompt when the computer or browser sees the webaccess in the wrong zone.
IE settings look like this.
so you can see if you come in on the internet zone the login details are not passed.
How would I go about turning on the authentication diagnostics?
My Network Admins are looking into the IE settings as they are managed via GPO. Thanks for the recommendation.
The IE Settings are the most likely culprit, but failing that - Go to control panel and check Credential Manager as the affected user and remove any related saved credentials. That could potentially cause this as well.