How high is the TLC5940?
All information on the PDIP NT package seems to have disappeared from TI's website. That package, and the PDIP version of the TLC5940 have been obsolete for years. It's no longer even mentioned on their website, and the latest data sheet, the one you link to, doesn't have it in the ordering information either. It's 2015, DIP is dying a slow death.

Based on @ThePhoton's link, you can see that the max height for a DIP, metal or plastic, is 4.57 MM. Most are lower though, 4 MM max. 0.1" headers come in various heights, but 0.1" is 2.54 MM as a reference. Use 4.57 MM as your reference height for the DIP when designing your stacking board height.

Alternatively, just use some calipers to measure the package once they arrive.

Found an older version with the DIP data intact. Data sheet from 2005. It's a quite large 0.2"/5.08mm body.