Anything is possible, but any modern system would detect a over heating issue and shut it self down in order to protect itself.
It's very possible that your graphics card is just about dead or seized bearings on the fan.
The thermal compound should not be "all over the cpu and gpu", you only need a very thin layer between the cpu, gpu and heat sinks.
When you applied the new paste how much did you use? You only need a very small amount, if it is squeezing out the sides of the gpu and cpu, then you are using too much, wipe it off immediately.
Make sure get a good quality thermal paste, I've used Arctic Silver and it works really well, not a fan of thermal grease.
Put in the new parts in and cross your fingers ;)