Most frequently seen problems with EM console have to do with non properly configured networking environments and skipping the dhcp warning at the installation phase. Proper networking configuration prerequisites have to be met before proceeding with installation.
1. DHCP issues.
Assigning a dynamic IP address makes the EM console to fail. Since the repository is configured with the environment it collects at install time, if a dynamic address is assigned, next time the computer unplugs (most frequently seen on laptop computers) EM stops working.
2. Use of IE with Enhanced Security Option enabled.
This is another issue frequently seen, when IE Enhanced Security option is enabled EM stops working, even though the console can be started.
3. Enterprise Manager can be accessed but Performance and maintenance tab request again for login credentials
This is another issue due to improperly configuring the networking and host name resolution environment. Oracle reads the hosts file to establish both the IP Address and the fully qualified host name. If it fails to properly read the hosts file, or if this doesn't properly identify the host, then the localhost.localdomain will be taken to configure the EM Console. When Attempting to solve the tns entry to access the performance tab or the maintenance tab Oracle won't be able to identify the host declared at the tns entry and it will fail to access, even though the provided credentials and the tns entry seem to be well configured.
4. Changing hostname or fixed IP Address
When networking changes happen then an EM reconfiguration is required.
For further references on HowTo troubleshoot and reconfigure Enterprise Manager I suggest to refer to this link --> Enterprise Manager DB Control Console.