Services fail to start with error "Cannot open database ArchiveManager requested by the login." (4216954)

After a reboot, the Archive Manager services do not start automatically. 

In the Event log, there are errors like:


The Exception message was: [Cannot open database "ArchiveManager" requested by the login. The login failed.
Login failed for user 'Example\ArchiveMgr_Service'.] and it occurred at:
System.Data.SqlClient.SqlException: Cannot open database "ArchiveManager" requested by the login. The login failed.
Login failed for user 'Example\ArchiveMgr_Service'.
at System.Data.SqlClient.SqlInternalConnection.OnError(SqlException exception, Boolean breakConnection)
at System.Data.SqlClient.TdsParser.ThrowExceptionAndWarning(TdsParserStateObject stateObj)
at System.Data.SqlClient.TdsParser.Run(RunBehavior runBehavior, SqlCommand cmdHandler, SqlDataReader dataStream, BulkCopySimpleResultSet bulkCopyHandler, TdsParserStateObject stateObj)
at System.Data.SqlClient.SqlInternalConnectionTds.CompleteLogin(Boolean enlistOK)
at System.Data.SqlClient.SqlInternalConnectionTds.AttemptOneLogin(ServerInfo serverInfo, String newPassword, Boolean ignoreSniOpenTimeout, Int64 timerExpire, SqlConnection owningObject)
at System.Data.SqlClient.SqlInternalConnectionTds.LoginNoFailover(String host, String newPassword, Boolean redirectedUserInstance, SqlConnection owningObject, SqlConnectionString connectionOptions, Int64 timerStart)
at System.Data.SqlClient.SqlInternalConnectionTds.OpenLoginEnlist(SqlConnection owningObject, SqlConnectionString connectionOptions, String newPassword, Boolean redirectedUserInstance)
at System.Data.SqlClient.SqlInternalConnectionTds..ctor(DbConnectionPoolIdentity identity, SqlConnectionString connectionOptions, Object providerInfo, String newPassword, SqlConnection owningObject, Boolean redirectedUserInstance)
at System.Data.SqlClient.SqlConnectionFactory.CreateConnection(DbConnectionOptions options, Object poolGroupProviderInfo, DbConnectionPool pool, DbConnection owni

System.Data.SqlClient.SqlException (0x80131904): Cannot open database "ArchiveManager" requested by the login. The login failed.