As a data point for people coming up on their interview, my letter said bring my passport OR birth certificate. I was asked 3 times for my birth certificate at the interview (even though I had my passport). I told everyone who asked me for it (check-in person, US CBP and Canadian CBP), the letter said OR, not AND. I was told that the letter is a known problem, and was asked to bring my birth cert with me the next time I rolled through YVR so they could "check it off" the list.
Thought it was interesting, I was approved and have my card without showing the BC, just good for everyone to know they really wanted to see a BC too...
*My processing time was quick, I already had GE, conditional approval was March 2016, interview was May 2016 and NEXUS card received 7 days after interview.