It doesn't matter if your using a Diecimila, NG, STK500 Dev Board, or just have your own Atmel in a circuit.
Unless you can observe the reset pin with a O-Scope, then we can't really solve this today.
However, it can be stated that if your circuit produces a characteristic on the reset line that brings the voltage down below 0.2*Vcc =1V and does so for >=2.5microseconds, then we can say the Atmel is being reset.
See the following image for reference,
Also, if your supply voltage goes below brown out voltage AND brown out detection is turned on, then your Atmel can reset. Does anyone know if the brown out fuses have been set on the Diecimila? on the NG? It would have been done with an in-circuit programmer.
If you can verify that the brown-out detection is turned on, then consult the following table when you find out at which voltage,Other questions which I'm not sure are answered (or completely answered):
1.) How much power is your motor taking to run (full speed RIGHT before shut off)?
2.) Are you overloading the voltage regulator (some linear regulators can have thermal shutoffs)?
3.) What are the O-scope plots between the following points: RESET - GND, and 5v-GND right near the Atmel when your motor is running AND right before your Atmel resets?
4.) Are you creating EMI that is propagating through your circuit causing havoc at multiple places?
5.) Did you try carefully wrapping the Arduino in some kind of metal shield for EMI protection?
It's not as straight forward as saying that one is more sensitive than the other.
Characterize FIRST how your electrically stressing your Arduino. THEN, apply the appropriate measures to fix the problem. Otherwise aren't we all just shooting from the hip with guesses to fix the problem? If we have the data, then we can make a correction.
All this talk has made me really interested in getting that motor that you used. I would love to characterize the circuit more. Where did you buy that exact motor? If you're unsure, then maybe we can play with this at the next Arduino hacklab?