Weierstrass' M-test says that the series of functions on some set $X$:
$$\sum_{n=1}^\infty f_n(x)$$
if $\forall n \in \mathbb{N}, \exists M_n$, \forall x\in X where $M_n \geq |f_n(x)|$, so the majorant series $\sum_{n=1}^\infty M_n$ converges, then the original series converges uniformly and absolutely for any $x\in X$.
But is the reverse true?
That is:
If I can prove that for some specific sequence of functions, any sequence of $M_n$ with $M_n\geq |f_n(x)|$, will have a divergent series $\sum_{n=1}^\infty M_n$.
Does that prove that the original sum is not uniformly convergent? can you give any counterexamples?
No, that will not prove that the original sum is not uniformly convergent. For instance, let $f_n(x)=\frac1n\chi_{[n,n+1)}$. Then $\sum_{n=1}^\infty f_n$ converges uniformly to $f\colon\Bbb R\longrightarrow\Bbb R$ defined by$$f(x)=\begin{cases}0&\text{ if }x<1\\1&\text{ if }x\in[1,2)\\\frac12&\text{ if }x\in[2,3)\\\vdots\end{cases}$$But $\sup f_n=\frac1n$ and $\sum_{n=1}^\infty\frac1n$ diverges. So, if, for each $n$, $M_n\geqslant\sup f_n$, the series $\sum_{n=1}^\infty M_n$ diverges too.