Are there finite lattices which are not order isomorphic to a sublattice of $\mathbb Z^n$? Consider the lattice $\mathbb Z^n$ for some finite $n$ ordered by $x\leq y \leftrightarrow \forall i: x_i \leq y_i$. I'm having difficulty thinking of all lattice which is not a sublattice of this.
For example, the diamond lattice is isomorphic to:
        (2,2,2)
      /    |     \
(1,0,0)  (0,1,0)  (0,0,1)
     \     |     /
        (0, 0, 0)

Are there counterexamples? If not, is there a name for the theorem that they are all isomorphic?

While not every finite lattice is a sublattice of $\mathbb Z^n$, as amrsa pointed out, it is true that every finite poset is a subposet of $\mathbb Z^n$.
Given a finite poset $P$ consider the map $L\colon P\to\mathbb Z^P$ given by
$$
L(p)_q =
\begin{cases}
1 & \text{if $q\le p$}, \\
0 & \text{otherwise}.
\end{cases}
$$
Then $L$ is an order-preserving and order-reflecting injection and hence $P\cong L(P)$ as posets.
Note that this map factors through the boolean lattice on $P$ by sending $p$ to its downset $p_\downarrow=\{\ q\ |\ q\le p\ \}$ which then gets sent to the characteristic function $L(p) = \chi_{p_\downarrow}$.
