The time successive sets are must is when your are increasing intensity by decreasing rest times. Decreased rest is one method of progressive overload. So whenever rest time is not part of your progressive overload strategy, you don't have to have successive sets.
One strategy is to pyramid down with reps and/or weight. Say you do a high volume program with maybe 5 sets of an exercise, after warmup you pick a weight where you can get 10 or 12 reps, then each set you rest 60 seconds (or whatever time you set based on progression), and do as many reps as you can. That number goes down each set. Alternatively, you can pick heavy weight to begin, and continue to reduce the weight each set as you maintain the number of reps. You can also lower the weight in small increments, so the reps also decrease in small increments.
Another way you can use successive sets is just to change up a hypertrophy program. It keeps your muscles fatigued and pumped, which is another stress for them to adapt to. You don't do it that way all the time though. Much of the time you want to maximize the progressive overload via adding weight and/or reps. That requires longer rest periods, and you can save time, thereby getting in more volume by working a different set of muscles while the first is recovering.
So as you can see, in most situations it's not going to matter which way you do it. It usually just comes down to what works better with the split you're doing, and personal preference. Like everything though, don't do it the same way all the time.