The PHB says "When you take a reaction, you cant take another one until the start of your next turn"
In 5e "turn is NOT" equal to round.
So on round 3, on init 25 Sauron throws a magic missle spell at Chewbacca.
On that moment Chewbacca casts shield as a reaction on round 3 init w5,
Then when its Chewbacca turn on round e init 22, Chewbacca amazingly, awesomly and without consulting the rulebook at all reaches "the start of his nect turn."
After Chewbacca spends his turn on a "hairball" issue we choose not to delve into, later in round 3 on init 16 Chewing his bacca sees his close and person friend, Abe Lincoln of Vulcan, fall off the soapbox he was stsndong on and uses a reaction to casr feather fall.
And so, not only is Abe Lincoln saved but Chewing used two reactions on the same ROUND.
There are other cases of course, i think it can get to 3, maybe more, if an ability gives you a new TURN like say rogue thief 17 Reflex thingy.
What padt of the PHB that specifies round are you citing? I ask cuz i dont see it under REACTION or OA listing but since you said it was from the PHB i am certain it must be there, right?