Why does bleach feel slippery? My high-school chemistry teacher taught us the mantra "bases are slippery-soapy-slimy!" 

This question and this answer in Quora say this is due to saponification - conversion of lipids from the top layer dead skin cells into a soap-like substance.

During some household cleaning I ended up with a small amount of a fairly strong solution of bleach in water between my thumb and forefinger and noticed it felt quite slippery, and it took several seconds of rubbing under water for it to stop.

In this case I would guess that saponification is not the explanation for the very slippery feeling. Is there another explanation?
Actually it is saponification.  Bleach has alkali added to it, to stabilize it against decomposion to chlorine gas.

To wash your hands after contact with bleach was a wise move.