Daisy chaining a large number (20) of TLC5940 in serial. Bolstering signals?
First:
I noticed that each board has a 5V regulator for the LED driver. You state that your power supply is 5V too. This will mean that the regulator does nothing but cause a light voltage drop. The TLC5940 will work on a voltage slightly UNDER 5V. It has 3..5V operating range so that is no problem for the chip.
If your LEDs can stand it use a higher VCC voltage. It would be better if you e.g. use 6V and then you can loose 1 volt over the wire and the regulator.

Possible solution:
As it seems to work on 11 boards I would suggest you make a small 'signal repeater' board. Just a set of buffers which take the input signals and send them out again. Make e.g. five or six of those boards and insert them at regular intervals. (Do not think: it worked with 11 so 28/11 = ~3 boards. You will need a safety margin). I had a quick look and the 74HC7014 looks good: six non-inverting buffer with Schmidt trigger input. Probably good up to 1MHz at 5V. Maybe you can find the 14-DIP version which is easy to solder on a breadboard.

Last:
Next time you embark on an electronic project this size talk to an experienced electronics engineer. Preferable an old fart like me. We know the pitfalls for constructions like this. 



Your connection diagram looks better except for one detail: You are feeding the HCT clock driver boards from the VCC which is 6V. But your expansion boards have a regulator which make them 5V.  Try to find a 5V signal on one of the the expansion board adjacent to the clock driver and connect that to the HCT board instead of VCC. The alternative would have been to add a 5V regulator to each HCT buffer board. 

You should buffer all the signals. That way they all get more or less the same delay and thus as a group there is little change. In this case it works as all signals go in the same direction. There is no 'return' channel. 

You do not need to terminate the power/ground. As to terminating the serial signals. It is not  bad idea but as the system worked before with ~10-ish boards I would first try without. You can always add them. 
It would be nice to have a scope picture of the signals at the begin and end.