The referee does call “a foul and a miss”, but that is not necessarily the same as the case being discussed. A “miss” is often called when the player is snookered on the object ball, but in this case there is no “three misses and you’re out” rule. If you’re snookered, you get as many attempts as you want.
If, however, as stated above, you are free to hit an object ball directly but fail to do so three times in a row, then you concede the frame.
E.g. in the 2003 Regal Welsh tournament, a new record was set for consecutive misses in the 12th frame as Ian McCulloch snookered Dave Harold behind the green. Harold missed the reds 12 successive times, giving away 48 points in fouls.