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."?
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.