Why are some people so afraid to say 'I don't know'?

Ego. Protecting one's pride and reputation.

It's about social conditioning. You're generally more comfortable saying "I don't know" to friends, or saying it to superficial issues.
You're also very likely to hear "I don't know... but it doesn't matter / concern me." Or "but *excuse*" if they do admit they don't know.

People who thinking highly of themselves, or lowly of others, are likely to avoid placing themselves on a perceived lower status. They feel that admitting ignorance makes then seem stupid to the one asking the question.

Try this next time and see what happens:
Deliberately ask a question that will stump the person you are asking. However, before you ask the question, condition them in two ways:

1) Stroke their ego, and then challenge them with the question. "You know, I hear you talking about cars all the time and how you're can go toe to toe with the auto mechanics. I overheard this the other day at Midas and I'm going to make you put your money where your mouth is."

2) Instead of challenging them, ask the question to seek more knowledge. "Hey you know a little more about cars than I do, and I have an obscure question that I can't seem to find the answer to. This may be a little out there, and if you can even point me in the right direction, I'd be really appreciate it."

Chances are people will respond to 1) more defensively (ie they won't say I don't know, and are more likely to give a generic answer, or justify by saying "well I don't know your car." Something that will take the blame off of them for not knowing.
The ones responding to 2) will be more open to saying "Idk."

The reason I introduce this is because 1) is the extreme example of why people are unwilling to say "I don't know." They see it as a personal attack on their knowledge and intellect.

People are so rationally irrational, it's weird.
There are many reasons.

I can agree and personally associate with Adam Sanford's answer to Why are some people so afraid to say "I don't know"?

Another reason I've become familiar with, such as when people fall silent when confronted with "difficult" questions is that people like to put-on or give-off a certain impression, and, saying "I don't know," would undermine that impression.

Can you imagine saying you're a highly experienced (and successful) accountant, having moved-up in several positions through several companies. Most of the time you've done Financial Accounting, but, you apply for and accept a CFO's position in a company using (Job) Cost Accounting. At that level, it really doesn't look good to say: "I don't know."

Some people just never become comfortable with being able to admit fault, or that they don't know everything. "I don't know" is often interpreted as others as being of fault, or showing weakness (the same as some people are unable to apologize despite knowing they've done something wrong, or they wouldn't want something they've done to another done to them). Some people (type A personalities), just have to be right -- there is no such thing as "not knowing," and the answer is to talk in circles, detract, make excuses, etc.

There are a variety of reasons, and, they all have to deal with an individual's personal psychology and life experience(s).
Arrogance and pride.  They don't want to look uneducated or ignorant, so they bluff through the situation.  Unfortunately, you cannot fool everyone all the time, so it often backfires and looks like what it is: immaturity. 

It is impossible for anyone to know everything about everything, so it's only reasonable to sometimes find the need to say, 'Sorry, I don't know that, but it sounds interesting, so I will look it up!'

It sounds more intelligent to me than some hem and haw bluffed up answer.
Because saying "I don't know" does two things:
 First, it gives the person who said that the impression he's not as smart as he thought. And of course that we dislike that, we like to believe that we know a lot of things, that we are capable of doing anything in this world and we know everything that can be known.
 And second, if someone says "I don't know", the next thing he should do is to learn to do that thing, or to learn that information, or, better said, to do something about their non existent knowledge.
 So people will always try to avoid that expression, they will try to seem like they know everything because they are scared to admit that they, as humans, have no idea about something and because we are so lazy that we are not capable to make something productive and learn new things.
Thanks for the A2A

I never claim absolute knowledge and say I don't know all the time, when I don't know. I also don't know anyone that seems to have this problem so my answer is - I don't know.

However, I do know someone who gets extremely frustrated and will simply stop talking when she doesn't know, after she admits she doesn't know. The reason is she mistakes ignorance with stupidity. She thinks she should know everything even though she hasn't been exposed to it. I always try to reassure her that ignorance of a subject is nothing to be ashamed of. If she is really interested in knowing about the subject start studying it. How helpful am I? - I don't know.
It requires a simple, open mind to admit it. Most of the people will not admit so openly because they are afraid of their image getting spoilt and their knowledge and abilities will be appraised poorly.
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