Why are my pieces breaking? "3612 Arm piece straight with 2 and 3 fingers"
During the 90's I was gifted older LEGO bricks from a friend of my parents that were about 30 years old. Many were discolored, warped and specifically the blue and gray pieces would break if flexed.  

30 years forward and some of pieces from my 90's mix are discolored specifically gray, blue, white, yellow, brown.  I think there is shelf life of about 30 years with certain color blends.  Note that pieces that have sat in sunlight have been affected more.  Maybe the sun's UV rays are not so good for certain color blends.

A few suggestions:  

(1) Buy new.  Head over to Bricklink and buy new pieces.  Unfortunately, I've never seen this piece in Green and the site is saying that this is somewhat rare. (Site wise only two exist! Sold by a store in Belgium at $2.16 each. Ouch!).

Record keeping purposes.  After breaking out my pieces from the 90's, I have creations and designs that once disassembled would be lost.  I don't want to lose these old drafts, or the build knowledge gleaned from them that I could use in future creations. Things can get more exasperated if these pieces are breaking due to time.  

(2a) Get a virtual LEGO program (I still use the discontinued LEGO Digital Designer) and preserve your builds digitally.  Label them correctly and store them indefinitely in The Cloud.  

(2b) For me, I can look at a model and figure out how it was made, so it might be faster to use your phone and make a short video of your creation from all angles, and upload that for preservation to The Cloud.

And finally, those pieces I have that are at least 60 years old, a few blue pieces can be crumpled in one hand...