Edit regarding 3+1 spacetimes and causality
I'll keep adding to the answer as I get more information, and hopefully everything will just evolve along. At the very least, I'll have a set of notes to work from in the future :) This is also the first, broadest, cut at an actual answer regarding causality.
Alcubierre sets out to find his warp drive metric using a 3+1 formulation of spacetime. In the 3+1 formulation, spacetime is described as a set of constant coordinate time spacelike hypersurfaces, (foliations, for the fancy). In doing this, you wind up with a line element that looks like (see erudite comments from @Jerry Schirmer below, I'm playing catchup):
$ds^2 = -d\tau^2 = \gamma_{ij}dx^idx^j + 2\beta_i dx^i dt - \left(\alpha^2 - \beta_i\beta^i\right)dt^2$,
where $\alpha$ is the lapse function, and is positive, and $\beta$ is the shift vector between spatial foliations. $\alpha$ describes how quickly time evolves, while $\beta$ describes how the spatial coordinates evolve in time. In other words $\alpha$ and $\beta$ describe how your ship moves through space and time per incremental step.
What's important here is that $ds^2$ is positive and for real space, $\gamma_{ij}$ is as well. Remember, hyperbolas look like $\dfrac{x^2}{a^2} - \dfrac{t^2}{b^2} = 1$. So, the line element equation above describes a globally hyperbolic system in space time. What's that mean? It means you can't close a curve in spacetime, so you can't violate causality. Note that $\beta^i$ squares up where it's important to maintain sign to maintain a hyperbola. I'd think there should be another requirement that $\alpha^2 > \beta_i\beta^i$, but Alcubierre doesn't mention this, so I'm guessing we don't actually need it.
Alcubierre isn't done yet, he's still got to find a metric that will fit in a 3+1 spacetime and do what he wants, (provide faster than light propulsion), but if he does, the above property of 3+1 spacetimes will guarantee causality.
Edit
I Stand Corrected Regarding the Alcubierre Drive
@Superbest pointed out, that the claims for the drive were that it could go faster than the speed of light with regard to the laboratory frame, and hence with laboratory velocity. I found the original paper by Alcubierre on arxiv[2], and...
he's absolutely right!
The paper is amazingly well written and folks that have had a grad level general relativity class should be able to easily traipse through it. Alcubierre even shows that causality won't be violated. I haven't had time to digest the material enough to say why causality isn't violated except with the very unsatisfying statement, "Well, the math works out." Alcubierre was also quick to point out that he felt that with a bit of effort he could come up with an example that would violate causality:
"As a final comment, I will just mention the fact that even though the spacetime described by the metric (8) is globally hyperbolic, and hence contains no closed causal curves, it is probably not very difficult to construct a spacetime that does contain such
curves using a similar idea to the one presented here."
OK, so to summarize. The math explanation and associated formulas I wrote below are correct. With uniform acceleration and no exotic matter whatsoever, you can travel more than x light years in x proper time years. In the case of the Alcubierre drive, however, that's not the trick they're playing. I hope to have more details soon, but in the meantime I'll leave you with this quote from Schild regarding the twin paradox and general relativity.
"A good many physicists believe that this paradox can only be resolved by the general theory of relativity. They find great comfort in this, because they don't know any general relativity and feel that they don't have to worry about the problem until they decide to learn general relativity."
End Edit
The explanation given in the Washington post article triggers a pretty common misconception:
"If an object reaches a distance x light years away in under x years, then it must be travelling faster than the speed of light."
What the article failed to mention is that the 14 days quoted is in the reference frame of the ship. The equation for the distance travelled with respect to time in the frame of the ship, (known as proper time), is
$$\mathrm{distance} = \dfrac{c^2}{a}\cosh\left(\dfrac{at}{c}\right)-\dfrac{c^2}{a},$$
where $a$ is the acceleration of the ship and $c$ is the speed of light.
Using this formula, it can be shown that at an acceleration of 188g, (188 times the acceleration due to gravity), the ship could reach Alpha Centauri in 14 days of ship time. You might point out that 188 g's would surely smush everyone against the back wall of the ship, but the beauty of the theoretical drive described is that you carry your own gravity well along with you and therefore, you're always in freefall and don't feel the acceleration.
Here's the problem though. The time that will have elapsed here on Earth will be much, much greater than the 14 days that elapsed on the ship. The expression for the time elapsed on Earth is
$$\mathrm{Earth\ time\ elapsed}= \dfrac{c}{a}\cosh\left(\dfrac{at}{c}\right),$$
which can be used to show that when the ship reaches Alpha Centauri, 817 years will have passed here on Earth.
The calculations shown here are nothing new, by the way. Rindler applied them to the problem of relativistic space travel for the first time in 1960 in a Physical Review article titled "Hyperbolic Motion in Curved Space Time" [1].
References
Rindler, W., "Hyperbolic Motion in Curved Space Time", Phys. Rev. 119 2082-2089 (1960).
Alcubierre's original warp drive paper
http://arxiv.org/abs/gr-qc/0009013v1
There is currently no theoretical upper limit to the Alcubierre drive. That being said, let's delve a bit deeper into it.
The Alcubierre drive, at its core, is just a deformation added on flat spacetime as a bump function (it is everywhere flat except for a compact region of spacetime). This deformation moves around, and there are no upper speeds dictated by general relativity.
The limitations will come from whatever matter field is used to prop up this metric. As is well known, it is negative (it will violate the null energy condition). While there are forms of matter that will violate the null energy condition, those are usually pretty small violations. To be precise, the energy is roughly of the order of
$$E = - \frac{v^2 c^4 R^2 \sigma}{G} \approx - (1.21 \times 10^{44}) v^2 R^2 \sigma$$
$v$ the speed of the warp bubble, $R$ the size of the warp bubble and $\sigma$ the inverse of the wall's width. There are ways to bring down those energies slightly, but basically that's the kind of energy scale we're dealing with. Even with extremely low speeds and tiny bubbles, the energy requirements are still pretty large.
It's pretty hard to find exact limitations for the speed, since that would involve proving a fair many theorems, but it's likely that we can't have the energy too negative. There are no precise generic theorems but various energy conditions and quantum inequalities points to the fact that there will be a bound on negative energy, and from what we can produce currently (via squeezed vacuum states, Casimir effects and such) and current theorems, it is likely to be rather small.
Those are the things to consider for the speed limitations of the Alcubierre metric. No clear answer, but even a particle-sized warp drive going at a slow pace of 1m/s is already gonna be pretty intense, energy-wise.
Best Answer
It's very, very highly likely that none of these schemes are workable for the following reasons: