If anyone's interested, TI themselves seem to have decent stock of the TLC5940. I just got 6 as samples due to me mistakenly ordering 4 chips after having already ordered 2 (and forgotten about them).
I only intended to take a small sample order, because what I really need for my final design are the 32-pin QFN version, but I can't fit QFN on a breadboard for testing, now can I?
Digikey has the PDIP version in stock at 511 count for $3.50 each if you're interested in buying.
As for my samples, I certainly won't complain about having too many. I'm likely to break a few considering how clumsy I am, so it's nice to have replacements. ;D