This answer elaborates on Willie Wong's comment , as well as providing and also provides another class of examples. Start with a clique $K_n$, pick two vertices $u, v \in K_n$, and glue two triangles onto $K_n$ at $u$ and $v$. It is easy to see that for any $n$, this graph is not Hamiltonian but there do exist exactly four pairs of vertices that are the endpoints of a Hamiltonian path. Furthermore, instead of using $K_n$, any subgraph such that there exists a Hamiltonian path between the distinguished vertices $u$ and $v$ will still do the trick.
This answer elaborates on Willie Wong's comment, as well as providing another class of examples. Start with a clique $K_n$, pick two vertices $u, v \in K_n$, and glue two triangles onto $K_n$ at $u$ and $v$. It is easy to see that for any $n$, this graph is not Hamiltonian but there do exist exactly four pairs of vertices that are the endpoints of a Hamiltonian path. Furthermore, instead of using $K_n$, any subgraph such that there exists a Hamiltonian path between the distinguished vertices $u$ and $v$ will still do the trick.