This is a common problem when computers are off the network for a time. The computer account password is periodically reset unless it is configured not do so. (See the security option Domain member: disable machine account password age.) The problem can also be with the server where the files are stored. If it is not connected to the network or otherwise having its own domain logon issues. On the other hand, if you are if you are sure that offline files is working as expected and all files are synching properly. If the only problem is to remove the balloon reminders, you can prevent them by adding the "turn off reminder balloons" policy in Group Policy. You can find the setting at Computer Configuration, Administrative Templates, Network, and Offline Files. The setting is also available in Explorer on the Tools menu, folder options and offline files tab. However, if it is set in Group Policy you cannot adjust it in explorer.
This was first published in February 2004