Has anything changed with you system... new duct work or new filters? Have you had an electrical storm before this problem started? Has the system always operated as expected until this problem?
If possible, the next time the motor stops, go to the furnace, open the blower compartment and CAREFULLY put you hand on the motor. Is it very hot? Do you smell anything that might be from excessive heat? Spin the blower wheel. Does it spin freely and quietly? If the motor is excessively hot then the thermal safety may be shutting it off, as previously suggested. However if that's the case, then you'd expect to find an excessive load on the motor (depending on the type of motor) for some reason... closed supply vents, blocked returns, clogged filters or a blower wheel that does not spin freely.
It would be nice to measure the voltage at the furnace control board where the thermostat wires connect between the G (fan) terminal and the C (common) when the blower stops. If the voltage is 0 then you know it's the thermostat. If you are reasonably handy with wire, you might try the following. This may be risky so think the logic through before attempting. I do not know your specific system or how it has been wired.
Turn OFF the power to the furnace. At the control board, disconnect the wire from the thermostst going to the G terminal. Label that wire G and put tape over the bare end. Now disconnect the wire from the thermostat going to the R terminal of the control board. Again label and tape it. Be sure you get the thermostat wire in case there is more than 1 wire at the R terminal. You have now removed power going to the thermostat and isolated the fan circut from the thermostat.
Now make a jumper wire and connect it at the control board from the R terminal to the G terminal. You are effectively putting 25V AC directly to the fan circuit thereby having continuous call for the fan and no thermostat intervention. See if the blower motor cuts out as it had before. If it does then the problem is with the control board/relay or the motor.
Testing the motor can be done but is more involved. You'll need to know about the kind of motor it is (PSC or other) and possibly make a direct connect to the motor bypassing the control circuit completely. This is not something most people would want to attempt.