Fat cells are called adipocytes. They are called lots of other things too. Usually nothing too complimentary. When a person gains weight, do they increase the size or increase the number of fat cells they have? Or both?
Once you reach adulthood, the evidence suggests that the fat cell number is the same in both lean and overweight people, so when you gain weight as an adult you are just cramming more fat into every adipocyte such that they enlarge.
Conversely, weight loss in adulthood reduces the size, but not the number of fat cells. Even those that have undergone surgery to reduce stomach size (gastric stapling) retain the same number of cells storing fat despite considerable weight loss.
The number of adipocytes is determined during childhood
Storing extra energy as fat has always been important for human existence. Fat cells became storage packs for reserve energy to help us get through any food shortages we may have encountered. Without the ability to store fat we might be able to live only a day or two without food. With adipocytes we have a much longer opportunity to obtain food.
If you cannot change the number of fat cells during the adult years, then any difference in cell number must happen during childhood and adolescence. Although there is speculation as to when during our youth we can increase the number, it seems that adolescence is a key time of accumulating additional fat cells. You can see why there has been concern about overweight children - three quarters of them become overweight adults. Developing an excess of fat storage cells during childhood may be one contributing aspect to chunky adults.
Today's fat is different to the fat you had in 1999
About 10% of your adipocytes are renewed every year. Old ones die and are replaced by fresh ones. The average lifespan of a single fat cell is around 8.5 years. Their ability to fill with fat is enormous. Someone at 140 kg (308 lb) having the same number of fat cells as someone at 80 kg (176 lb) just shows you how "efficient" they are at doing their job.
What does it all mean?
The adult body has around 80 billion adipocytes. They are there for life, being lost and replaced slowly. You can't change the number of fadipocytes you have at your 18th birthday. You can, however, have a big say in their size. The paper did not answer the question whether liposuction permanently reduced adipocyte number, although they stressed that there is a "tight regulation of adipocyte number" through life.
Note: this is my interpretation of the data given in the reference (Nature 2008; 453: 783-787)