...though unfortunately this is widely believed. When the M16 was first used in Vietnam, it was assumed that the smaller 5.56mm round would make much smaller wounds than the 7.62mm M80 round fired from the M14. Everyone was surprised to learn that M16 wounds were often much more severe. In order to explain this discrepancy, it was theorized that the slow 1:14 barrel twist made the bullet less stable in flesh and caused it to tumble, resulting in the large wounds. In fact, the slow twist only made the bullet less stable in air. Any pointed, lead core bullet has the center of gravity aft of the center of the projectile and will, after a certain distance of penetration, rotate (yaw) 180° and continue base-first. This is where the appearance of "tumbling" came from.
The actual cause of the larger-than-expected wounds was not a result of this yawing of the bullet, but of the velocity of the bullet coupled with the bullet's construction. M193 bullets have a groove or knurl around the middle, called a cannelure. This allows the mouth of the case to be crimped on to the bullet, preventing the bullet from being pushed back into the case during handling and feeding. The cannelure also weakens the integrity of the bullet jacket.
When the bullet struck flesh at a high-enough velocity, the bullet's thin jacket, weakened by the cannelure, could not survive the pressure of moving sideways through the dense flesh. Instead, the bullet would only rotate about 90°, at which point the stresses were too much for the bullet jacket and the bullet would fragment. The results were a wound that was far out of proportion to the size of the bullet. Yet, the twist rate of the barrel and therefore the rotation speed of the bullet, is not a factor in the fragmenting equation.
M855 ammo works exactly the same way, though due to its heavier bullet, it has less muzzle velocity. Less muzzle velocity translates to a shorter range in which the bullet retains enough velocity to fragment, compared to M193.