Why do the motors for my Arduino RC car not respond to any command after the first command is received sometimes?
To add to my comment, it sounds like the Arduino is suffering a brownout condition. I think the correct term is actually called voltage sag but essentially it means a temporary drop in voltage due to high current draw and the internal resistance of the battery.

It's likely the batteries you are using are not able to deliver the instantaneous current required to drive the motors. What happens is when the motors are first switched on, they initially draw a very large current and causes the battery output voltage to dip. After a certain point, the Arduino senses a brown-out condition and shuts down.

Depending on how bad the brownout is, you might be able to get away by throwing on a huge capacitor close to the input of your Arduino (after the voltage regulator which I presume you have). Hopefully that will tide over the initial high current draw.

Otherwise, you can try having separate power sources. One dedicated to the Arduino and the other dedicated to driving the motors. I haven't looked into your motor drivers but this configuration is supported by most drivers I've seen and used. They'll usually have two power inputs on the drivers: one for logic level voltage input and one for power input.

If you want to share a single power source, consider using a battery that is able to deliver high currents. You can look on websites like HobbyKing for lithium-ion batteries that can deliver high currents. Do read up on how to handle li-ion batteries if you decide to go this route though.

Without seeing your circuit or any oscilloscope pictures of the power rail, there's not much else I can suggest at this point.