Do shows steal stories from other shows?

A reader/Sitcom Room attendee, Johnny Walker, posed an interesting question:

I was recently surprised to see a storyline I recognized in an episode of How I Met Your Mother: In the 2009 episode "Double Date", Marshall reveals that he can only fantasize about other women in his head, if he imagines a tragic illness has taken Lily away from him first. (He feels too guilty, even in his dreams, to