OK, here goes. The High Priests (Kohanim) are descended from Moses' brother Aaron. In order to be a Kohan (Cohen, Katz, and all similar names), you need to be the son of a Kohan, and also be pure of mind and spirit. One of the scourges of Biblical times was leprosy. Before leprosy is evident as skin lesions, it often affects one of the nerves to the hand muscles (the ulnar nerve). Thus, someone who couldn't make the proper handshape for the blessings was likely not pure of body, and may be harboring leprosy. They therefore could not fulfill the requirements of being Kohan.
I just love how ancient rituals often have a medical explanation. Call me weird...