So, with some snooping of our own, we managed to fix the problem.
I reset the stored credentials for the user account and the Exchange Management Console is working fine again.
I found the answer in this post:
http://www.petri.co.il/forums/showthread.php?p=118602The directions it suggests are:
1. Click Start, click Run, type control keymgr.dll , and the click OK.
2. Click Back up, and then follow the instructions to save the current entries.
3. Delete any entry that matches the names of the Exchange servers and domain controllers in your organization. Typically, you want to clear all entries.
4. Try to open the Exchange Management Console
Thank you to everyone that helped me with this problem.