Why does one counterexample disprove a conjecture?

Can't a conjecture be correct about most solutions except maybe a family of solutions?

For example, a few centuries ago it was widely believed that ${2}^{{2}^{n}}+1$ is a prime number for any n . For n=0 we get 3 , for n=1 we get 5 , for n=2 we get 17 , for n=3 we get 257 , but for n=4 it was too difficult to find if this was a prime, until Euler was able to find a factor of it. It seems like this conjecture stopped after that.

But what if this conjecture isn't true only when n satisfies a certain equation, or when n is a power of 2$\ge $ 4 , or something like that? Did anybody bother to check? I am not asking about this conjecture specifically, but as to why we consider one counterexample as proof that a conjecture is totally wrong.P.S. Andre Nicolas pointed out that Euler found a factor when n=5, not 4 .

Can't a conjecture be correct about most solutions except maybe a family of solutions?

For example, a few centuries ago it was widely believed that ${2}^{{2}^{n}}+1$ is a prime number for any n . For n=0 we get 3 , for n=1 we get 5 , for n=2 we get 17 , for n=3 we get 257 , but for n=4 it was too difficult to find if this was a prime, until Euler was able to find a factor of it. It seems like this conjecture stopped after that.

But what if this conjecture isn't true only when n satisfies a certain equation, or when n is a power of 2$\ge $ 4 , or something like that? Did anybody bother to check? I am not asking about this conjecture specifically, but as to why we consider one counterexample as proof that a conjecture is totally wrong.P.S. Andre Nicolas pointed out that Euler found a factor when n=5, not 4 .