For a DC motor with a stall current of 950 mA, what should the H-bridge's current rating be? What will happen if we use our H-bridge L293D whose max. output current is 600 mA?
First a bit about motors. Stall current is the current drawn by the motor when the recommended voltage is applied and the motor is not turning due to a load. Alternatively, the no load speed is the speed the motor will spin at under no load. In this state it will draw a minimum current.
Here is an example torque/speed current for a DC motor:
So whether or not you can use your H-Bridge depends on how you are using the motor. If you are using the motor at a high speed and thus low torque state, you will not approach the stall current and may stay within the spec for the motor driver.
Furthermore, torque/speed curve is voltage dependent. If you operate at a lower voltage, you will draw less current and could possibly stay within spec for the H-bridge.
If you exceed the recommended current draw for the H-Bridge, it will heat up and possibly burn out or in the worst case start a fire. The L293D has overtemp protection so it may just stop functioning.
There are ways to increase the max current capacity of the H-bridge by introducing cooling in the form of a heat sink or fan or liquid cooling, etc. The current through the H-bridge is causing resistive heating of the chip. The heating raises the temperature of the chip eventually causing damage. If you increase the flow of heat out of the chip, you will be able to heat it more (more current) without raising the temperature too high.
Another option is use two L293D chips in parallel (H-bridge piggybacking). Adafruit suggests this with their motor arduino shield which uses L293D chips. They state this will double the current capacity. http://learn.adafruit.com/adafruit-motor-shield/power-requirements
Other H-bridges you could checkout are the L298 and DRV8801.
What Brendan said.
However, there is an out if you want to reach for it: once you have determined the actual stall current for your situation, from the motor armature resistance, supply voltage, and the whopping 1.4V drop in the L293, if you pulse the drive to the motor such that the average current through the L293 is the rated 600mA or less, then it shouldn't burn up.
You'll need to pay attention to switching speed, and where the current goes on the back side of the PWM, and all those other pesky details, and if your PWM generator stalls at the wrong portion of the cycle things will get toasty, but that doesn't mean there isn't a path to success with that part.
(Personally, I'd figure out how to deal with surface-mount parts and put in the DRV8801 -- it looks far superior. Or, if I never needed to generate top torque at top speed, and if efficiency wasn't a concern, I'd just put a resistor in series with the motor).