Doing it fast will potentially cause the sagging skin. Most people can only lose weight in a healthy way, at around 4lbs of fat a week max and you've got to be really morbidly obese to do that.
For most people around 100lbs or less overweight, 2lbs a week is a sensible target to aim for, though expect a bit more than that for the first 3-4 weeks, especially if you are consuming significantly less sodium in your diet, as some of the weight loss could be excess retained water, but then the weight loss would hopefully plateau out to a sensible steady amount.
Losing weight too quickly, I.E. crash dieting, like say 8lbs a week for example, could cause this, as the skin would struggle to recontour, to the ever decreasing proportions of the body.
However, if the person was really overweight and had caused significant amounts of elastin damage, then sagging skin would almost certainly be inevitable, even if they were achieving sensible weight loss amounts per week, as without enough structurally sound elastin, the ability for the skin to bounce back would be compromised.
When loose skin is an eventual inevitability, the only real option thereafter is surgical restructuring I'm afraid.