Scratch that statement: Hypothetical modes of hyperspeed travel are exactly as realistic as time travel. If you have any method which can be used to get you from point A to point B before a photon travelling through vacuum could get there, the same method could be used to travel into the past, and vice versa. Note that I am not qualifying “any method”, there. It could be hyperspace, it could be wormholes, it could be warp drive, it could be teleporting dragons. It doesn’t matter.
There are a number of theoretical methods to achieve faster-than-light travel (and hence time travel). They’re understood, in the sense that we can write down meaningful equations for them. But all of the known methods (and probably all methods, period, though I’m not entirely confident on that) rely on having exotic matter, which has negative mass (that’s the oversimplified version; the actual condition required is somewhat more mathematical). Unfortunately, there’s no evidence for the existance of negative mass, and no reason beyond wishful thinking to even think that it might exist.
No, it’s far worse than that. Even centuries ago, people knew of things which could break the sound barrier: Bullets and whips come to mind. The question then wasn’t whether it was theoretically possible for things to break the sound barrier; if there was a question at all, it was merely whether it would be practical to build something to break the sound barrier carrying a person. Note that there’s a huge difference between “possible” and “practical”.
This brings us back to the question of starships which are slower than light, but still very fast. There’s nothing inherent to the laws of physics which prohibits travelling at, say, .9c . It’s perfectly possible, for instance, to travel any distance you like at a constant acceleration of 1 g (fast enough to make a round-trip to nearby stars in a single lifetime), even just using chemical propulsion (you’d just need a ludicrous amount of fuel). This is a problem in the same category as previous generations speculating about supersonic travel: We know it’s theoretically possible; the question is just whether there’s any way to make it practical. If one’s measure of “practical” is generation starships, then sure, it’s doable, but the OP seems to want a bit more. As a rough answer, I’d say that in order to reach other stars in a single lifetime with a plausible amount of fuel, we’d probably need some way of converting most or all of the mass of our fuel to energy (fusion, at a fraction of a percent, isn’t efficient enough). This is not a completely insurmountable obstacle, the way that negative matter seems to be: I know of at least three methods within the known laws of physics to convert a significant fraction (half or more) of a mass into energy. I hasten to add that all three are far beyond the reach of our current technology, and we’ll probably reach the stars by slower means before we develop any of them. But it’s possible.