A good strategy to find examples that break this bound is to use Xi's construction of the *dual extension algebra*, which keeps the number of vertices, doubles the number of arrows, and exactly doubles the global dimension:

<cite authors="Xi, Changchang">_Xi, Changchang_, [**Global dimensions of dual extension algebras**](http://dx.doi.org/10.1007/BF02567802), Manuscr. Math. 88, No. 1, 25-31 (1995). [ZBL0851.16003](https://zbmath.org/?q=an:0851.16003).</cite> 

For instance if the quiver is an oriented cycle ($n=m$), then the maximal finite global dimension that can be attained is $2n-2$, see

<cite authors="Gustafson, William H.">_Gustafson, William H._, [**Global dimension in serial rings**](http://dx.doi.org/10.1016/0021-8693(85)90069-9), J. Algebra 97, 14-16 (1985). [ZBL0571.16011](https://zbmath.org/?q=an:0571.16011).</cite>

Since $\mathrm{gldim} (A)=2n -2 < 2n=n_A+m_A$, such an algebra $A$ with maximal finite global dimension is not itself a counterexample. But for its dual extension algebra $D$, we get $\mathrm{gldim} (D)=4n -4$ and $n_D+m_D=n + 2n = 3n$. So for $n \geq 5$, $$\mathrm{gldim} (D) >n_D+m_D,$$ and the bound does not hold.