Such practices are commonly applied to results of some random number generators, but less often to other applications.
There is one very common such fail-safe test
NIST's FIPS 140-2 requires very simple contiguous test for random number generation: store previous generated block and compare. Only if the block is different it is assumed that the RNG did not fail.
From FIPS 140-2 4.9.2:
Continuous random number generator test. If a cryptographic module employs Approved or non- Approved RNGs in an Approved mode of operation, the module shall perform the following continuous random number generator test on each RNG that tests for failure to a constant value.
More entropy tests
The standard (FIPS 140-2) used to require more tests (mono bit test, poker test, runs test, long runs test). These tests give fairly good estimate of if input appears random. These were decommissioned, because they work better with NDRBG than RNGs required by the standard (see FIPS 140-2 Annex C if you need details), and because for some cases it is excessive requirement to produce 20000 bits for analysis (it may be that you just needed one 128-bit key). AIS31 standard used to evaluate true random number generators uses some very similar tests.
Used in practice
These tests are useful against some hardware random bit generators and for this reason, Linux rngd program is running the tests (previously part of FIPS 140-2) to ensure quality of entropy gotten (for example from /dev/hwrng) and to ignore inputs which appear non-random.
You may use Linux rngtest program to apply these tests to data of your choice, for instance, examine output of (possibly) buggy encryption software.