Not really - in most laptops the main processor is cooled by the forced air from the small fan on the processor itself, taking the heat from the finned copper heatsink and directing the hot air out through the vent on the bottom of the case. Most overheating occurrences arise either when the fan itself fails, or when the vents are completely blocked with dust accumulated over time.
The heatsink and fan assembly are not really open to other parts nearby, so I doubt that the breakage of USB port could have disrupted the airflow unless somehow the broken parts jammed the fan and stopped it from turning. Even then, failure would not be immediate. The build up of heat would have been obvious, however, and the laptop would likely have stopped functioning if it got too hot.
The fact that you had a failure on switch-on suggests to me that something else may have occurred - it is not really indicative of overheating unless you were aware that the laptop was excessively hot before it was switched off.
In your case the USB port breakage may have had side-effects you were not aware of - such as metal or copper print shards being disturbed by movement, then falling into sensistive parts of the motherboard, damaging the machine on switch-on for instance.
This is just speculation on my part, but I know of one case where a dropped laptop worked perfectly well for weeks. The owner ignored a rattle coming from the laptop, until one day it failed to boot up. This was the day that the rattle bit back - it was really a very small metal spring that had broken loose from the DVD drive. It had moved around once too many times, and had finally shorted components in the motherboard. End of laptop.
-Stewart
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