Thank you @JoeN for taking the time to reply. Here's a picture of my output circuit:
In theory, the 2K ohm IREF resistor I have on my TLC5940 sets the current sink of the LED outputs to 20mA for each of the 16 channels, so that's where the 20mA/channel * 16 channels = 320mA comes from.
RPU in the above image is 470ohms, and VLED is 12V. So I believe the voltage at the output pin of the TLC5940 is 12V - (20mA * 470 ohms) = 2.6V. I would expect power to be sunk from all outputs would then be: 16 channels * 2.6V * 20mA = 832mW. This is well below the rating of the chip, and with just a single chip connected this all runs fine.
When I connect my second TLC5940 PCB in series with my first is when I have a problem. I suspect the second TLC5940 is sinking additional current from the first TLC5940 PCB. Here's a picture of my TLC5940 PCB schematic. Perhaps you can spot something that would be causing a current drain form the first board thru the 2nd TLC5940.