What would cause SRAM to have no errors when 5 of the bits are high?
Signal integrity? 

What is the length of your ribbon cable? A long cable will cause
your beautiful square signal coming out the driver looking bad at the
load.

And what is clock rate of the transmission into the cable? Fast transmission might induce signal integrity issues such as cross-talk between the lines.
Try to add a GND line in between every signal in your ribbon cable.
It's the speed that tells you if your cable is long or not. (and more precisely, it's the rising/falling time of the signal, not its toggling frequency... )
Power integrity?
If you have only one small wire for Vcc, this could create a voltage drop that is big enough to disturb your chip.
If you have only one GND for the power and all the signals, this might cause the same issue.
Did you put enough decoupling caps at the power pins of you chip? (both chips)

As @Ben Jackson suggested: Is Vcc properly connected?


And last (but not least):


Software: Are you sure about the code you use to check your hardware? You might be looking for a hardware bug, when it's your test software that has a bug.


This is just guesses, but related to my experience, it's highly probable that the issue is here somewhere.