What firmware are you using?
Most firmware has an option to turn off the extruder if the hot end gets too hot. A lot of people turn this off at the start when testing as it also affects things like not having a thermistor or thermocouple connected at all.
Check the signal on Arduino pin D10 to see if it is constantly on when this happens. If it is not, then you most likely have a bad MOSFET.
Q: Where did you get your board from, and what MOSFETs are on the board (Note: They may not all be the same time).
Note: Some vendors put a different MOSFET on the boards than the one in the schematic. In some cases they choose a suitable replacement (the design is old, that component is harder to get, and newer better components now exist), but sometimes the replacement they choose is not very good and can fail like this. If you buy a replacement MOSFET yourself, I recommend something like the IRLB8743 (also IRLU8743), which is used on many other RepRap electronics variants. Another similarly suited replacement is the FDP8870, which is still a better performer than the original MOSFETs, but slightly less capable as the IRLB8743.