There are no rules that require it.
This answers the question; feel free to stop here.
Following is a list of reasons why requiring it is a bad idea.
What is normal?
Think about the real people you interact with in the real world. If they could be represented by 6 numerical statistics generated by rolling 3d6 (+1 for being human) then 90% of them would have scores between 7 and 16. Normal is a very broad area; exceptional is very small.
If PCs use a non-rolling method the absolute lowest score than can have as a human is 9; 8 for variant humans and other races. So, think of a random group of 20 people you know. Every D&D PC is smarter, wiser, stronger etc. than at least 3 of those 20 people. That is, the dumbest PC is smarter than a lot of the people you interact with on a daily basis.
How much is luck?
If you choose to make a PC take a skill check, how much is luck, how much is attribute and how much is experience?
Well, barring magic, a human PC can have a stat of between 9 and 20 giving a modifier of -1 to +5; a range of 7. Their proficiency bonus can range from 0 to 6; a range of 7. By the rules, natural talent (ability scores) is given equal weighting to experience (proficiency). Someone who is an expert (i.e. has expertise) gets twice as much benefit from experience.
However, luck as represented by the roll of a d20 has a range of 20 so it is nearly three times more important than talent or skill.
What do the scores represent?
It is absurd to think that 6 numerical scores can in any way represent a person. They are game constructs and exist only for resolving the mechanical aspects of the game.
Let's take intelligence for example. Does it represent an ability to:
- solve algebra?
- follow a philosophical argument?
- recall who won the 1956 World Series?
- get yourself elected President?
- determine the optimal time to substitute players in a football match?
- determine from an X-ray if someone has lung cancer?
- decide who wins a court case?
- design a plumbing system?
- install a plumbing system from someone else's design?
The obvious answer is yes it does but can a person with 20 intelligence do all of this; clearly not. Can a person with 6 intelligence do none of this; also clearly not.
What if the boot is on the other foot?
If you require your players to play down to their scores, how are you going to enable them to play up?
That is, if they have a character that is smarter, wiser or more personable than they are how to you get them to role player being more e.g. insightful than they are in real life?
Who are you to judge?
What guidelines are you going to use to decide that a player is playing outside their character's capabilities? How can this be anything but arbitrary and capricious?