All the nonsense word fluency measures is the ability of the child to be able to read v-c-v combinations. By the time the students reach the end of first grade, almost all of them have this test down pat.
For kindergartners, DIBELS (stands for Dynamic Indicators of Basic Early Literacy Skills, a University of Oregon-created measuring program) employs sound segmentation to measure whether the student can recognize and break up sound segments in words, the v-c-v patterns in the nonsense word fluency, letter recognition, and first-sound recognition. Yeah, these five- and six-year-old kids are going to be scarred for life going through four minutes of torture. For first graders, from the middle of the school year onward they are "tortured" with the nonsense word fluency and with reading three passages. Second graders have the nonsense word fluency in the fall but then on have the three reading passages to measure fluency. Third graders through six graders have the three reading passages, plus they have the so-called DAZE passages, which measure reading comprehension and are better indicators than retells. These probes are hardly anything to get a stroke over.
DIBELS is used like AIMSWEB in other school districts to measure student progress in reading (and in AIMSWEB, math as well). I am sure there are similar programs in other school districts that I am not aware of.
In my experience giving these probes, I don't see a lot of anxiety in students over these assessments, by the way. They are used to them. In the district where I work, these probes are administered three times a year.
Now CRTs and standardized tests, that's another ball of wax.