Ablative Body Armor
We all think that in "real warfare" it's one shot one kill, but for the past few decades, that have been becoming less and less true. Many armies now equip thier soldiers with body armor that includes ceramic plates that can stop most battlefield ammunition. These ceramic plates are typically rated as class-IV body armor meaning that they can stop pretty much any kind of handgun, submachine gun, assault rifle, LMG, and most sniper rifles even if they are using "armor-piercing" bullets. But once they are hit they shatter making them useless against follow-up hits.
And that is just where tech got to about 20 years ago. Ablative body armor is still getting better. Another recent development is the use of shear-hardening polymers that are used to treat modern kevlar body armor. SHP treated kevlar becomes so hard under impact that it becomes like a ceramic plate. SHPs make kevlar about 4x as strong for thier weight, but like all hard things, they break instead of bending. While the SHP coating itself re-liquifies after being hit, allowing the armor the take multiple strikes, the underlying kevlar can be sheared from impact weakening the armor as a whole.
Apart from armor getting lighter for its stopping power, armies are beginning to equip thier infantry with robotic exo-frames. These robotic suits allow a soldier to carry several times as much weight as he could unaugmented meaning that in the very near future one can expect infantry armor to more closely resembled that of a light tank or APC.
With the kind of military grade body armor you will see in the not so distant future, not only is your description possible, but it seems inevitable. Unless someone figures out how to make a much better bullet, the best way to take out such a target would be to riddle it with high caliber bullets shattering the ceramic plates and shearing the kevlar fibers, until the armor becomes weakened enough to penetrate or you need to nail it with a direct hit from an anti-vehicle scale weapon like an RPG.
The reason people still use assault weapons in this scenario would be the same reason they still work in video games. RPGs are big and heavy, and only one-shot on a direct hit. The cost of ammo in terms of both manufacturing and weight make giving everyone RPGs impractical; so, most troops are still given automatic weapons because in the 2-3 seconds it takes to line up a shot to make good and sure your RPG will hit, an assault trooper could spray and pray 20-30 shots into someone's gut waring his way through the armor for the kill.