Most examples of ample line bundles that are not globally generated have less number of global sections than the dimension of the variety. Assuming ampleness, is the existence of "enough" global sections sufficient to guarantee globally generated-ness? More precisely:
Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $n \ge 2$ and $L$ an ample invertible sheaf on $X$ with $h^0(L) \ge n+1$. Is $L$ globally generated?
The answer is no for curves, and then a product construction show that the answer is no in every dimension.
Take a hyperelliptic curve $X$ of genus $g \geq 3$ and let $P$ be a 2-torsion line bundle on $X$, such that $P = \mathcal{O}_C(p-q)$, where $p$, $q$ are distinct Weierstrass points. Now set $L = \omega_C \otimes P$.
Since $L$ has strictly positive degree on a curve, it is ample. Moreover, by Serre duality, $h^1(C, \, L)=h^0(C, \, P)=0$, hence $$h^0(C, \, L)=g-1 >2$$ by Riemann-Roch. On the other hand, we also have $$h^1(C, \, L (-p))=h^1(C, \, \omega_C (-q)) = h^0(C, \, \mathcal{O}_C(q))=1,$$ hence $$h^0(C, \, L(-p))=1+(1-g)+ (2g-3)=g-1.$$ But then $|L|=|L-p|$, in other words, $p$ is in the base locus of $L$ and so $L$ is not globally generated.
Finally, take two copies $(C_1, \, L_1)$, $(C_2, \, L_2)$ of the polarized pair $(C, \, L)$ and consider the line bundle on $C_1 \times C_2$ given by $$\mathscr{L}=L_1 \boxtimes L_2=p_1^* L_1 \otimes p_2^*L_2.$$ This is ample by Nakai-Moishezon, since it has positive self-intersection and intersects positively every effective curve in $C \times C$. By the Künneth formula, we have $$H(C_1 \times C_2, \, \mathscr{L})=H^0(C_1, \, L_1) \otimes H^0(C_2, \, L_2)$$ and so a basis of sections for $\mathscr{L}$ is given by elements of the form $\sigma_1 \boxtimes \sigma_2$, with $\sigma_i \in H^0(C_i, \, L_i)$. But all these elements vanish at the point $(p, \, p) \in C_1\times C_2$, hence $\mathscr{L}$ is not globally generated.
This provides a counterexample to your question in dimension $n=2$, and similar constructions provide counterexamples in every dimension. Note that, by increasing the genus of $C$, we can make the integer $h^0(\mathscr{L})$ arbitrary large.
