Multiple TLC5940s on PCB "dropping out" when many outputs turned on
It is obvious that your power distribution needs beefing up. 

Your ground plane looks OK, but I'm very (as in, "deeply, deeply") suspicious of your VCC distribution. You're using a skinny little trace with a single small via to provide VCC to each group of 4 ICs, and that just won't work. Worst case you're trying to pull about 4 amps through each trace, and that's just not in the cards. Worse, the middle segment of your distribution trace will need 8 amps.

The cleanest solution is to go to a 4-layer board, and devote one layer to VCC and one to ground, with the other 2 for signals. Assuming you don't want to do this for cost reasons, I'd recommend running a 24 ga wire from the VCC input to each IC. At a minimum, one wire for each pair (for instance, BLUE1 and BLUE2), with a jumper between adjacent ICs.

Your quick check should be a simple voltage measurement from your ground test point at upper left to the IC VCCs while under load. Any voltage droop of more than about 0.1 volts should be a red flag.