The words "durchfallen" ("to fail") and "der Durchfall" ("diarrhea") seem to have completely different meanings. Is it possible to use the former to mean "to have diarrhea" and the latter to mean "failure"? Do listeners associate between the words, e.g. giggle when you say "Ich bin durchgefallen."?
Failure at school is "das Durchfallen". (e.g: Die Regierung will das Durchfallen in der ersten Klasse abschaffen.)
The phrase "Ich bin durchgefallen." does not connotate diarrhea.
There are many informal words for diarrhea anyway.
I support thei's answer that "durchfallen" has no connotation with diarrhea and vice-versa. At the risk of sounding a bit disgusting here, the similarities are still there though.
With "durchfallen", you cannot keep up and fall through the cracks.
With "Durchfall", you're unable to control your feces, they just fall through.
So in my opinion, the description of falling through something is what actually links failing and diarrhea - they're using the same metaphor, stretching it a bit in different directions.
I would say that "Durchfall" is a "durchfallen" (failure) of a particular sort. That is, failure to hold your -----. At least that's how I remember them.
So they're really not separate. One is a special case of the other, more general term.