Sounds pretty fast... you're tapping @ ~42 sfm. I looked up what Balax recommends - they recommend 20 sfm in 300 series SS - although they do not specify any difference between 303 and 316, I think I'd dorp it down to about half of where you're at. If it's a blind hole, you will want to drop another 25-50% in sfm. Make sure your coolant concentration is high, like 10-12%, or maybe some Moly Dee is in order.
If you are not using a thread forming tap, try one...they work great and the cold formed threads are really nice - just finished a job with about seventy 10-24 threads in T316, used a Balax "Thredfloer" tap and first and last hole were perfect (dribbled Moly Dee in every blind hole just to make sure).
If your doing a through hole, and don't want to stand there dribbling tapping fluid, you can program a move into a small cup full of fluid off to the side of the table to pick up fluid, and then go thread...with a thread forming tap instead of a cutting tap, you don't have to worry about chips either.
Steve