I'm looking for examples of how certain kinds of "systems" based on
Fibonacci numbers can be shown to be amazingly accurate.

For example, (this is MADE UP, please don't waste your time correcting
this error-filled example):

"The gyroscope in the Mars Explorer used Fibonacci techniques to land
the ship within 500 feet of it's destination - after a journey of over
5 million miles!"

The best examples will be predictive - and for well-known or broadly
used technology, not some obscure niche product or device.

How about US Patent (6,281,681
(http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=6281681)
images link: http://patimg2.uspto.gov/.piw?Docid=06281681&idkey=NONE)
to GE for an MRI reads: ?Rapid acquisition of NMR image data from
which an image can be reconstructed is achieved in an MRI system
employing a predetermined pulse sequence to acquire NMR data that
sample k-space in a Fibonacci spiral trajectory. The pulse sequence
may be repeated to sample along a plurality of interleaved Fibonacci
spiral trajectories, and if the number of interleaves is set equal to
a number in a Fibonacci series, the total sampling is itself a
Fibonacci spiral which samples substantially uniformly over k-space.?

Basically, the virtue of employing the Fibonacci quasi random
numbering is increased speed of the imaging process hence reducing
adverse effects from the human moving that sort of thing.  I know the
example is quite a bit off from the nomenclature you seemed to desire,
sorry.
 
There are also uses in signature verification so on and so forth.
Michael