Your theory about an overheat cutout sounds plausible - a thermocouple generates electricity depending on the temperature at its tip - used to measure jet pipe temperatures etc.
You are probably thinking of a thermostat - a simple heat-operated switch which is what many (perhaps all) vacuum cleaner motors have.
To test your theory you might consider something a bit off-the-wall. Operate the machine from room temperature and time the period to cut off.
Some time later - wrap it up in polythene and put it in the fridge - if you can make it fit - for an hour - then see if it runs for longer.
Some time later warm the machine up with a hair dryer and then run it - if it cuts off sooner then you have a probable diagnosis.
If you go to this page....
http://www.dyson.co.uk/store/parts.asp?accessory=ACC-DC0714MOTOR&product=DC07-SY
You will see this text...
Motor 240v YDK £36.31
Not available to order online.
If you think that the motor needs to be replaced, please contact the Dyson helpline on 0800 298 0298.
It may be that the safety cut out thermostat is an intregal part of the motor...
It's a freephone number so you have nothing to lose...
Good luck...
Peccavi
June 2010