The National Institutes of Justice armor ratings are set forth in
Ballistic Resistance of Body Armor - NIJ Standard-0101.06. The specification for Level IIIA soft armor is as follows:
Type IIIA (.357 SIG; .44 Magnum)
Type IIIA armor that is new and unworn shall be tested with .357 SIG FMJ Flat Nose
(FN) bullets with a specified mass of 8.1 g (125 gr) and a velocity of 448 m/s ± 9.1 m/s (1470ft/s ± 30 ft/s) and with .44 Magnum Semi Jacketed Hollow Point (SJHP) bullets with a specified mass of 15.6 g (240 gr) and a velocity of 436 m/s ± 9.1 m/s (1430 ft/s ± 30 ft/s).
Type IIIA armor that has been conditioned shall be tested with .357 SIG FMJ FN bullets
with a specified mass of 8.1 g (125 gr) and a velocity of 430 m/s ± 9.1 m/s (1410 ft/s ± 30 ft/s) and with .44 Magnum SJHP bullets with a specified mass of 15.6 g (240 gr) and a velocity of 408 m/s ± 9.1 m/s (1340 ft/s ± 30 ft/s).
So it is rated to stop a 125-grain 0.355" bullet at 1470 ft/sec, or a 240-grain 0.429" bullet at 1430 ft/sec, both of fairly low sectional density. IIIA is known to stop some bullets that are smaller and faster (e.g. commercial 5.7x28mm out of a pistol length barrel), but not
that much faster due to the physical limits of Kevlar. Almost any centerfire rifle cartridge designed since 1894 will exceed those parameters by a
very wide margin.
Pistol rounds are relatively low-pressure as firearms cartridges go, and they don't typically have much barrel length to make the best use of what pressure they have to work with, so pistol (and most revolver) rounds tend to be relatively slow (Mach 1.4 or less) and low energy (relatively few handgun rounds reach even 600 ft-lb of energy at the muzzle). As a result, you can stop a handgun round by catching it in a mesh of lightweight, strong polymer fibers that stretch and break as they decelerate the bullet within a couple inches or so. Rifle rounds have too much energy for that; even the lowly .223 Remington exceeds 1200 ft-lb and 2800 ft/sec out of a 16" barrel, and most deer rifles carry between 2000-3000 ft-lb, and you can't decelerate a projectile with that much velocity and energy with any known fibers within the space of a couple inches. (If you could let the net deform a foot or two, it'd be easy, but a vest can't deform that much without grievously injuring the wearer.) Instead, you have to cause the bullet to expend its energy disintegrating itself against something harder than itself (e.g. Level III rated AR500 plate), or dissipate its energy crushing and pulverizing an extremely hard ceramic (Level IV plate, the only practical way to stop tungsten or tungsten carbide core AP rounds).
The gist of the armor ratings are as follows:
NIJ Level IIA (soft armor) - rated to stop slow 9mm, .40 Smith & Wesson (handgun rounds)
NIJ Level II (soft armor) - rated to stop fast 9mm, slow .357 (handgun rounds)
NIJ Level IIIA (soft armor) - rated to stop .357 Sig, .44 mag (handgun rounds)
NIJ Level III (hard armor) - rated to stop steel jacketed 147gr 7.62x51mm at 2780 ft/sec (substantial rifle round)
NIJ Level IV (hard armor) - rated to stop full power rifle shooting armor piercing ammunition (166gr .30-06/7.62x63mm tungsten-core AP at 2880 ft/sec)
For context, M855 is a 62-grain, copper jacketed, steel-capped lead core .22-caliber bullet at about 2980 ft/sec out of a 20" barrel.