Once the control rods are in, the reactor is losing more energy than it is producing and will eventually cool down. However in many old (e.g. 60's era) designs

Once the control rods are in, the reactor is losing more energy than it is producing and will eventually cool down. However in many old (e.g. 60's era) designs including Chernobyl and Fukushima, the reactor is losing that heat via active water cooling. If you turn off those heat pumps, then the core is still energy-positive even with rods inserted. In Chernobyl the core literally blew up, so it's a different situation. But in Fukushima the pumps lost power and so even though the control rods were fully inserted, there was a real risk it might get hot enough to melt down anyway. That was thankfully avoided.