As other answers have explained, if you have a claim that something is true for all possible inputs, then a single counterexample disproves the claim. Period.
Perhaps, though, you may have some lingering doubt about how "generic" the counterexample is. Sure, maybe we can prove that $\lvert a+b \rvert = \lvert a \rvert + \lvert b \rvert$ is false for $a=(1,4,5)$ and $b=(2,2,2)$. However, for all we know, this might be the only counterexample. We already know the claim is false (beyond any doubt), but maybe it's only "technically" false. Maybe it's true "in the typical case", and we just happened to find the only exception.
For example, suppose I claim "$x^4 + y^4 \ne z^4$ for all integers $x$, $y$, and $z$". You say, "That's not true: $0^4 + 1^4 = 1^4$". You'd be right: Indeed my claim is incorrect. But if you're willing to let me move the goal posts a little, I can easily salvage the claim by excluding "trivial" counterexamples like $0^4 + 1^4 = 1^4$. The claim remains true "in the typical case" - "technically" false but true "in spirit".
Most of the time you won't see people explicitly address how "generic" their counterexample is. They'll just give one counterexample and be done. And implicitly, they're saying that this one counterexample is convincing enough that the claim is "generally" false. Because:
When a statement is false, it's "generally" false... generally.
This belief is completely non-rigorous and not formally justified, but that's okay - there was no formal meaning for "typical case", "generally true/false", and "trivial" to begin with.
How could we make a more convincing argument that a statement is not only false, but "generally" false?
Depending on the situation, we might try to show that:
- There are infinitely many counterexamples (not just one)
- There are infinitely many counterexamples that aren't just obvious
variations of each other (e.g. not just constant multiples of each other)
- The set of counterexamples has infinite volume
- A random input (drawn from a chosen probability distribution) is a counterexample with probability $1$
... the list goes on. In the most fortunate case, we can find a definite answer by describing exactly which inputs satisfy the claim and which ones don't. For example, it turns out that "$\lvert a+b \rvert = \lvert a \rvert + \lvert b \rvert$" is true if and only if $a$ and $b$ are positive scalar multiples of each other, or one or both of them is $0$. This is a rigorous statement which is stronger than merely disproving "$\lvert a+b \rvert = \lvert a \rvert + \lvert b \rvert$ for all $a$ and $b$", and can be informally interpreted as saying that "$\lvert a+b \rvert = \lvert a \rvert + \lvert b \rvert$" is "usually" false.
These are the ways you could go beyond proving that a claim is false, to prove that it's "very" false.
But enough talk about false statements being "generally false" or "only technically false". It's still false. All such talk involves moving the goal posts, replacing the original statement with a different one.
For whatever actual, specific claim you might be considering, one counterexample is enough to disprove it.