There are two types of testing, and I'm not certain what your question refers to.
On the one hand, wire and cables are tested and then 'labeled,' meaning an NRTL such as
UL. Most industrial wire and cable has also been tested by the Canadian Standards Association, but part of NAFTA was an agreement that the CSA and UL standards be harmonized. On that UL page, one of interest may be
type 1063, a good one to look at. Your application may use several types, I'm not too familiar with many and anyway UL is currently revising several standards. The actual tests performed are generally written in the UL standard for that product (something you're expected to buy) and the tests themselves are written in ASTM or ANSI documents (something else you're supposed to buy.) These tests (such as flammablility) agree with the listing requirements for connectors, cable ties, etc. Very frequently, the tests performed are mentioned or even partially explained in manufacturer data, such as the Belden/ Manhattan/ Atlas etc. catalogs.
In terms of performing tests to customer specifications, these tests can be added to. It is routine for even a moderate user to specify additional tests when a specialized product is made. Once the cable hits the dock, there are three tests performed which I've seen, which means it is not an all inclusive list. By far, high pot testing is most prevalent. Many cable houses (custom cable assembly shops) will also perform a point to point check using rather fancy cable testing machinery, similar to a bed of nails PCB test. A third test, which is applied mainly to purchased item cable assemblies, is the pull test. But a pull test is for connectors, which would be a different subject.