Graphs which can connect every pair of vertices via a hamilton path

Few years over date, but given that the answer by Mateusz seems wrong: these graphs are named Hamiltonian-connected graphs and some sufficient conditions can be found in section 3.2 of this dissertation. The variants on Dirac's, Ore's, and Posa's theorems are a bit stricter.

Update: For example the equivalent of Dirac's theorem is: If G is a graph of size $n\geq3$ such that $\sigma \geq \frac{n+1}{2}$, then $G$ is Hamiltonian connected. (Where $\sigma \geq \frac{n}{2}$ is enough for Hamiltonicity).