Interesting, owo. Might have to go do that soon..
I have good news and bad news. The good news is, it's not the wires! Yay!
The not-so-good news is I found out by connecting the chip directly to the Arduino. Using the same wires that control the first four chips just fine, I put them to the fifth chip, and it didn't work.
I know from experience that any chip I put there won't work, so the breadboard must be at fault. I have no idea how or why -- I used a multimeter to test all the strips and they work fine. I didn't have to stick the probes in far (they didn't even fit into the holes), so it doesn't make much sense to me.. Though the chips do pop up, maybe it has to do with that even if they're stuck down.
Regardless, point is the issue can be tracked down to me having a faulty breadboard. Is there any way to fix this or should I just grab a new one?