Debugging charlie-plexed LED array

All of the online tutorials that mention charlie-plexing and LED array over using shift registers list pros and cons. The pro is always few components, simpler layout etc. and the cons always include "difficult to debug".

They're not wrong!
Since getting our 90-LED charlie-plexed array soldered up nearly a week ago, there's still little improvement on getting it to work properly! We did track down a few broken traces (the black toner in the printer was a bit faint in places so a few traces over-etched and had breaks in them). But even after fixing those, and spending hours and hours with the continuity tester/multimeter, there are still six different scenarios that cause the matrix to fail.

Although we feel we're close to solving it (having identified which pins cause it to fail) the last bit - understanding our findings - is still as elusive as ever! Here's the array working as it should:

The sheet in the background is a drawing of our findings - the six scenarios that cause the matrix to fail and the LED pattern created, instead of a single LED lighting up


Every now and again, however, we get something like this result

 The LED we expected to be lit up is in the centre of the "cross" shape - and remains stubbornly unlit

Even more peculiarly, we did some measuring with the voltmeter. When an LED is lit properly (from a 5v source running through 200 ohms) the voltage across the cathode and anode pins is around 2.9V. The voltage across all the other LEDs in the same row/columns is about 0.6V which is the typical voltage drop across a diode, but not enough to cause it to light up.

When we measured the voltage across the pins of an LED that wouldn't light up (the one we expected to light up but is in the middle of the cross - and remains unlit) we got almost a full 5v across the two pins. All the other LEDs that were faintly lit measured about 2.8-2.9V. So in this case, with the dim LED in the centre of the cross, we're measuring 5V across the pins, but it does not light up.

Here's a diagram of the pins combinations which don't work, and the LED "patterns" that are lit up.
If anyone can spot a pattern or - better still - come up with an explanation, please leave a comment below (and anyone else in future having similar problems might have some idea where to begin debugging!)