I can't reproduce the problem. You'll be glad to know that the only change made between Arduino AVR Boards 1.6.20 and 1.6.21 is to support the new versions of the official Arduino Nano board. If you don't own one of those boards then there is no need to install Arduino AVR Boards 1.6.21.
Vaialos:
I go into %appdata%\Local\Arduino15\staging\packages and I delete all the files in there so that we're hopefully not trying to process old/broken garbage.
Good thinking!
If you do want to pursue a solution further I recommend you to post here the more detailed error output you will find in the black console window at the bottom of the Arduino IDE window after encountering the CRC error and then closing Boards Manager. Maybe that will provide a clue for someone here.