Maybe, but probably not.
The surprising thing is that the equations of Einstein gravity have solutions in which a closed path can encounter itself at an earlier time. So our best theory of gravity does permit solutions in which time travel is possible, but that's different from saying that time travel is actually possible in our universe.
It may be, as many suspect, that some other physics principle prevents such solutions from corresponding to the real world. After all, we expect Einstein gravity to break down at high enough energies because of quantum mechanics, so the theory of quantum gravity may render the time-travel solutions unphysical. However, there is also some work that suggests that quantum mechanics can cure some of the usual objections to time travel, such as the "grandparent paradox".
So we can't really answer the title question until we have a theory of quantum gravity, which may give hope to aspiring time travellers.
"Perhaps the biggest surprise of the work of the past decade is that it is not obvious that the laws of physics forbid time travel. It is increasingly clear that the question may not be settled until scientists develop an adequate theory of quantum gravity."