Connecting to SQL Server on same domain, but getting "Untrusted Domain" error?

We have multiple IIS web applications running on multiple web servers that connect to a SQL Server on a separate machine on the same domain.

Everything was running smoothly for over a year, then we had some major network issues. There was a problem with the replication between the primary and secondary domain controllers. There was an issue with Distributed File System services, Event ID 14530 ("DFS could not access its private data from the Active Directory.").

We got the network issues sorted out apparently, except now, none of the IIS web applications can connect to the SQL Server. We are using Windows Authentication. The error message is "Login failed. The login is from an untrusted domain and cannot be used with Windows authentication." This is weird because we have verified that both the web servers and the SQL Server are in fact on the domain.

To make things weirder, the Network Admin removed the SQL Server from the domain and rejoined it, and the websites worked again... but only for a few minutes. Then the same error started occurring again!

The Network Admin and I have been working on this issue for several days. Any ideas would be appreciated.

Edit:

We can connect using SSMS. Also, applications that do not use IIS can connect. For example, we have a developer with some Fox and Cloud applications that use connection strings that can connect.

We have tried removing and re-adding the web servers to the domain.

The errors in the SQL Server log are as follows:

Login failed. The login is from an untrusted domain and cannot be used with Windows authentication.

Error: 18452, Severity: 14, State: 1.

SSPI handshake failed with error code 0x80090311 while establishing a connection with integrated security; the connection has been closed.

Error: 17806, Severity: 20, State: 2.

Edit 2:

The system log on one of the web servers contained this error this morning:

This computer could not authenticate with [domain controller], a Windows domain controller for domain [domain name], and therefore this computer might deny logon requests. This inability to authenticate might be caused by another computer on the same network using the same name or the password for this computer account is not recognized. If this message appears again, contact your system administrator.

Edit 3:

I have tried pretty much everything suggested in this thread, including making sure the user was not locked out, the password was not expired, and the hosts file did not contain an invalid entry for localhost.