Suppose you want to show a series does not converge uniformly on some interval. If you know the point wise limit is $f$, and you can show the $\sup |f_{n} - f|$ does not go to zero on your interval, then that does it, the convergence can't be uniform.
Is this correct?
That is correct.
Uniform convergence implies pointwise convergence. So, given the fact that $f_n\rightarrow f$ pointwise as $n\rightarrow\infty$, the only possible uniform limit of $(f_n)$ is $f$.
But, as you say: uniform convergence of $f_n$ to $f$ on the set $X$ is equivalent to saying that $\sup_{x\in X}\lvert f_n(x)-f(x)\rvert\rightarrow0$ as $n\rightarrow\infty$. So, if this is not the case, then your convergence cannot be uniform.
In some cases, there are also other properties that you can look for; for instance, the uniform limit of continuous functions is continuous, and so if your $f_n$ are all continuous and your $f$ is not, then convergence cannot be uniform.