In the Zabbix v1.8.1 has been discovered that monitoring stopped for the monitored devices (SNMP devices) after shutting down these devices and after some time turning them on.
In the Zabbix frontend in the Configuration->Hosts these devices are shown as available (SNMP available in the "Availability" column), but in the Monitoring->Latest data report is visible that "Last check" for items of these devices are not updated since the time when devices were shut down.
For some unknown reasons in the Zabbix DB in the hosts.snmp_errors_from and hosts.snmp_disable_until fields remained old values, that is why monitoring were not working. After deleting these values monitoring started working again.