Of course this depends on the application.

There is no universal answer as there are infinite potential modes of failure of each of the zillions of parts used in electronics today.

I think it's quite appropriate for certain type of products to simply perform the smoke test, then configure (burn) and perform the "try to wiggle with all buttons" test.

If it's a board with "$2500's-worth of other components", in 100s series, there might be a JTAG-enabled (read boundary scan enabled, I know it's not the same nowadays, unfortunately) part and you might have developed some sort of testing via the JTAG. You need to perform the smoke test first, anyway; but this might save some hassle with shorted buses and similar stuff.

I'd not suppose such a bad part which would burn out other parts (except voltage regulators and alike, but that would be too bad if those would not be reliable enough, anyway - I'd thrash the supplier or manufacturer of such).

All this might be completely inappropriate for your needs and application though.

JW

PS. I've written a memory test when a "sclerotic" batch of SRAMs slipped through production to the field, but I knew exactly the mode how they failed from the examined specimen so the test was specific.