In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauss map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively).