This topic is kind of old but I'd like to share my information.
For workaround, I put a small low pass filter between the first SOUT and the second SIN, second SOUT and third SIN, and so on. This magically worked and flickering vanished out.
I succeeded in controlling several servos and LEDs with 10 TLC5940 serially daisy chained.
I am using TLC5940n DIP package.
Please look at the video in the post below.
Many people encountered the same problem: more than two ore three TLC5940 make the farther LEDs more flickering.
And they could not found any answers after all, as long as I crawled over several forums.
I had the same problem, and I wandered why no "big" project using lots of LEDs with TLC5940 exists.
Many applications are using WS2801 modules because its more reliable and widely spread.
I am not 100% sure, but I suspect the flickering problem everyone encounters stems from the "output timing" of the TLC5940.
This is not the voltage or any signal integration problem, but it comes from its specs.
As you know, TLC5940 accepts its SIN signal as SCLK signal triggers high.
Simply put, SIN signal should be already HIGH (or LOW) when SCLK gets high.
However, SOUT signal of the first TLC5940 changes its value almost SIMULTANEOUSLY as SCLK gets high.
This would be confusing the second and latter TLC5940's SINs and make the total mess.
I don't know this is the spec of the TLC5940 because TI does not tell this in their datasheet.
Please refer to the timing chart in モノを作りたくなるブログ: TLC5940 の SOUT の変なタイミング
Other problems may be coming from energy consumption of TLC5940.
TLC5940 draws fairly large power (5V,100mA, etc.) with careless programming implementation.
This causes instant drop of the voltage and makes flickering.
This voltage drop sometimes cannot be compensated by 0.1uF capacitor, especially when you are using a small voltage regulator.
At this time, you should monitor the voltage drop with your oscillo while you change the values of capacitors.
In my case it worked with 10uF and 47uF capacitors.
I hope this helps .