Hello everyone!
As many people over the forums and web have been affected by this issue after the Anniversary Update (basically when you try adding a PIN, spinning dots appear but nothing happens and the Add button is still there), I'd like to share a fix with all of you. As somebody may have thought, it has to do with the TPM (Trusted Platform Module) of your machine: it's a component that basically stores in a safe and encrypted way security keys, and is used by the whole Windows Hello thing. Anyway, it might be that the content of your TPM is corrupted and therefore needs to be resetted.
In order to do that, just type within the Cortana search box (or in cmd) tpm.msc, then click on the first result which will open up a new window about TPM management. On the right you'll have a column with a few options: click on Clear TPM. This will restart your machine and prompts you to a UEFI/BIOS message, which will ask you to press a specific button in order to clear the module, then it will reboot by itself again. When you'll be back in Windows, run again tpm.msc, but this time click on Initialize TPM: a few seconds and it will be done. Now you'll be able to add a PIN as you were previously to the Anniversary Update.
Hope this will be useful. Sorry for my bad english

Official stuff from Microsoft
Clear TPM: Clear the TPM
Initialize TPM: Initialize the TPM