Kay! Forgot this existed!
So, the decoupling capacitors didn't work. They helped a little but not enough.
Common sense struck. I wired the LED's to a different PSU than the IC's and now they work "fine". It fixed my problem but a new one came up, then I fixed that and... Well, long story short, things are worse now than they were before.
It all started when I added the second PSU -- I now have an ATX running the LED's that can drive up to 20 amps. Yeah, totally overkill, but at least I know I can turn them all on at once and nothing will blow up. Apparently I moved something around, because unknown to me, the 6th IC popped up /juuuust/ right so that five of its output pins and the Serial Outpit pin came out of the breadboard. This went unnoticed for a few days before I pushed it back in.
Doing that fixed "everything", except that when I toggled too many LED's on at once the 6th IC would turn all its outputs on. I have no idea why it did this but obviously this had to be fixed. I checked the wires, all good. No shorts anywhere, no components touching, nothing. There was no logical reason for it to be acting up..
..Then I had someone else look at it and we noticed that some of the 5th's LED's were acting up too. Checked the wiring and such, all good again. We figured that the IC had to be faulty so I swapped it out. Nope! Things got worse. Now none of the 5th's LED's light, but it gets weirder -- It's not broken. It's still getting serial data and passing it to the next chip in the chain. I even replaced it /again/ (this time with a chip from a new batch) and it did the exact same thing. Kinda.
The new IC listens, but it doesn't do exactly what I say all the time. If I only tell it to light one LED at a time, I can make all the LED's in the display go on and off in sequence and it works like a dream (almost). If I turn them all on at once, the ones controlled by the 5th IC are flaky. Usually they'll light but be very dim. If I bring the PWM duty cycle down sometimes they'll light as bright as all the other LED's, sometimes not. There seems to be no rhyme or reason to it.
Oddly, if I turn on individual LED's controlled vby the 5th IC, then turn all the LED's on, the ones I turned on remain fully bright while the others are dim. I can't even begin to fathom what's going on.
Worse still, now the LED's connected to the 7th chip won't respond at all anymore. They're always off. I don't think this is the 5th's fault, since the 6th still works absolutely perfectly. The 7th worked fine before but now it don't. Gah.
So, I'm stumped. I don't know where to start trying to fix things at this point. I suspect my house has been built over an Indian burial ground and I've been cursed with bad juju. What do I do?
EDIT: Forgot to mention that I did remember to tie the ground lines together, and didn't tie the 5 volt rails together. I might be stupid sometimes, but I know not to pump 20 amps into a laptop's USB port.