Try:
Settings / sound and display / audible touch tones to off.
Edit: my response is for Zahhak.ts
For the initial question, make sure your ringer volume is not all the way down, maybe?
I am unable to duplicate no tone with short tone selected.
Ah, very nice. Thank you much Todd. I never could figure how to turn it off on my BB Storm for the two yrs I had it. Now to figure out how to set text tones to a custom one I made with myxer.