I want to add a little bit of interpretation from my side. I don't think the time limits are directly connected to the engine overheat. The reason for the time limits is to ensure a reasonable engine life. Of course the time limits are indirectly connected with engine overheat as the temperatures will be higher with higher settings.
You may run the engine on "5 minute all out" setting for 1 hours without overheating, however this would reduce the time between overhauls considerably (maybe to 20 hours compared to 100 hours).
On the other hand you may run the engine on "30 minute climb" setting for 10 minutes and damage the engine due to overheating, because radiator was shut and speed was to slow.
So a "5 minute" limit doesn't mean it will only overheat after 5 minutes, it may overheat earlier and actions must be taken to prevent this. It also may overheat later or not at all and there won't be any trouble during this flight if the engine is new. There might be trouble in the next flight or you are grounded because of required engine overhaul