@dc42: Yes, I have. They run all the way.
@boffin1: I'm doing what you said (the second one, when it worked) -- Have been from the start, but thanks for at least telling me not to try daisy-chaining them.
@Graynomad: I tried another, now it's being even weirder. The chip works. I know this because it's responsible for eight green and eight blue LED's, and the eight green ones work flawlessely. However, the eight blue ones have gone nuts. They're very dim (normally they're the brightest on the display) and only turn on fully once in a blue moon (excuse the pun). However, if I turn only a few LED's on at a time, they work. I don't know what to think -- The PSU is more than capable of running every LED at once (it could run six of these displays at the same time), they're grounded, wires are in just fine.. I don't know what can be causing this. I even added a 6th IC to see if those blue ones wouild mess up too, and it just gets weirder. The next 11 do, the the last 5 are fine. I'm thinking about scrapping the project entirely and designing a PCB instead.
@dhenry: I've been doing that.