What we’ve got here is a failure to communicate.

Donn Pearce (1928-2017) American novelist, screenwriter

Cool Hand Luke (1967) [with Frank Pierson]
    (Source)

The line actually occurs twice in the movie. It's initially voiced by the Captain, the warden of the prison where Luke is sent to. While the line shows up in the final revision of the script (sc. 313) as above, Strother Martin actually speaks it in the movie without the indefinite article "a" before "failure":

     What we’ve got here is failure to communicate.

The second time the line shows up -- spoken by Luke, at the end of the movie, mocking the Captain -- it is both scripted (sc. 432) and filmed with the article intact.