'Ampleness' of a big line bundle

In my view being big corresponds to being "generically ample", that is being ample outside a closed subset. This closed subset is called the augmented base locus of $M$ and it is often denoted by $mathbb{B}_+(M)$ ( see http://de.arxiv.org/abs/math/0308116v2.pdf). So, if for example you take $M$ big and $F$ to be a line bundle you can prove that $M^motimes F$ is always globally generated (i.e. base point free) outside the augmented base locus of $M$ (note that you don't need global generation of $M$ for this). In the example of Anton Geraschenko in fact $mathbb{B}_+(M)$ is exactly the exceptional divisor. Something similar might hold if you consider locally free sheaves of higher rank.

Here is a simple counterexample (of the form Zhengyu Hu suggested). Take $X$ to be the blowup of a point in $mathbb P^2$, $M$ the pullback of $mathcal O(1)$ under the blowup map, and $F$ the line bundle associated to the exceptional divisor $E$. Sections of $Fotimes M^n$ are rational functions which may have a pole of order 1 along $E$ and a pole of order up to $n$ along a line not meeting $E$. However, any rational function $f$ actually having a pole along $E$ (i.e. generating $Fotimes M^n$ at the points of $E$) must have a pole along some other divisor passing through $E$ (since $X$ has the same rational functions as $mathbb P^2$).