This is both a philosophical question and a very important engineering application question.
In general, it is considered that "failure" in the context of a fatigue analysis means that you have sufficient load/cycles to initiate a crack. However, that does not mean that the component or equipment has failed.
My recommendation for in-service equipment is to take a fracture-mechanics approach to cyclic life. Postulate a crack size, based on the minimum-detectable crack size for the NDE inspection technique that you will use. Then, using the cyclic stresses from the cyclic loading that you calculated from your fatigue analysis, perform crack growth calculations. Calculate the number of cycles it will take for the crack to reach a critical size (usually based on the crack behaviour changing from ductile to brittle, but there may be other criteria, depending on the equipment/component). Halve that number, and that now becomes your inspection interval.
Every inspection that you perform that doesn't find a crack resets the inspection interval (and hence the life). However, expect Murphy's law to bite you in the a$$, so always have a repair plan in-place for when you do find a crack!
The problem with fatigue in general, and fatigue curve specifically, is that it is quite random. A graph of test data showing stress vs cycles will look like a shotgun blast at the paper. Most published curves will either be a lower-bound curve or a mean curve with a design margin to bring it down to approximate a lower-bound curve. Nevertheless, there is inevitably test data that falls below the curve, so at best you can think of these fatigue curves as -2σ or -3σ curves, with the expectation that fatigue crack initiation is possible.
Now, for welded structures, fatigue failure is likely to occur in the welds, as welds behave differently than smooth bar steel. Unless your fatigue analysis used something like the Structural Stress Method (Verity Method), you should almost expect pre-existing microcracks in welds to exist and have grown. Depending on the NDE originally performed on the welds, it is possible that the life of the welds is an order of magnitude or more lower than what you would otherwise expect for non-welded construction.