Weird. It's also working correctly for me now. For specifics, this is what was happening when I tried to block multiple different groups:
I'd click the block button and it would show the success message, but the group wasn't added to my block list and the block button wasn't changing into the unblock button. This kept happening no matter how many times I clicked the block button on various groups. After I went to my block list and removed a group, I was able to block a new group on the first try.
My best guess is that there was some problem with the site that affected block lists for a while. I guess I'll just wait a while and try later if it happens again.
Thanks for checking,
@YumGumYum