Some effects only appear in combination
Interaction, and why it is invisible to sequential testing.
Some effects only appear in combination because two factors can each be harmless on their own while producing a real problem, or a real benefit, only when both are present at once, a property of how the two interact that no sum of their separate effects would predict. This is the phenomenon a factorial design is built to catch, and seeing it in plain terms makes clear why it is worth the extra runs.
Tight laces on a steep hill
Shoelaces tied a little tighter than usual feel fine on a flat walk. A long downhill stretch in normally laced shoes is also unremarkable, tiring on the legs but nothing a foot would complain about. Put the two together and a blister can form that neither produced alone, because the tight lacing stops the foot sliding forward naturally inside the shoe on the descent, and the combination creates the friction. A test of tight laces on the flat and a test of the hill in normal laces would each report, honestly, that nothing went wrong. Only a walk combining both has any chance of revealing the problem, and that is exactly the combination a one-at-a-time study, testing each factor against an easy baseline, never tries.
When one factor changes how the other acts
An interaction exists whenever the effect of one factor depends on where a second factor is set. A factor with no interactions adds roughly the same effect wherever it is tested, so testing it alone against a single baseline gives a fair answer. An interacting factor might do almost nothing at one setting of its partner and dominate the result at another. A full factorial design catches this by measuring the first factor's effect at both settings of the second, and the difference between those two effects is the interaction, a number in its own right.
Interactions usually correspond to something physical, one factor changing the very mechanism by which the other acts, as the laces change how the foot moves on the hill. They can be as large as either factor's own effect, so a study reporting two individual effects without ever running the factors together can understate badly what happens when both occur at once.
A result that looks contradictory between two separate tests, a factor mattering a great deal in one context and barely at all in another, is often the interaction itself showing through. Treating that contradiction as a clue, where the temptation is to average it away as noise, is usually the fastest route to understanding what the system is doing.
Choosing which pairs to test
Not every pair that could interact does so by a meaningful amount, and hunting for interactions everywhere runs into the same combinatorial cost that makes full factorial studies expensive. The number of possible pairs grows roughly with the square of the number of factors, so ten factors already give forty-five pairs. The practical discipline is asking, before running anything, which pairs share a mechanism, act on the same surface, or feed the same downstream process closely enough for an interaction to be physically plausible, and reserving deliberate combination testing for those.