I read a quote from Hawking talking about how while time travel is theoretically possible, one should not be able to travel back in time to a point before the time machine was invented because there would be no time machine to be in/have you there. Following that logic, however, wouldn't time travel be impossible altogether since the moment you arrived at the time before you left, you would never have left?

Satori

That would make no sense if you could not travel to times before the time machine is built. The machine is not limited to the start of its existence since the machine does not control time, it maneuvers through it.

I'm going to have to point out that his theory doesn't take into consideration of the possibilities of tachyon particles/fields. Theoretically, if time travel is possible, it would create a tachyon field around the traveller/vessel essentially creating a "warp bubble". This "bubble" would protect the traveller/vessel from the effects of reversed time. The problem is, there is no real evidence that these particles exist yet...they are just theoretical particles. Plus, there is no way to tell IF a warp bubble would even be created, let alone If it was created, how long it would last before the traveller/vessel succumbed to reversed aging.