Using the speed of light in a vacuum or in that medium To clarify, is the speed limit of the universe the speed of light in a vacuum, or the speed of light in that particular medium, i.e. if the speed of light in a particular medium were only 17 m/s, would I be able to go faster than that speed? 
Also, for the special relativity equations of time dilation, I was always taught to use $c$, as in the speed of light in a vacuum. But to me, it makes more sense to use $c_m$ the speed of light in that particular medium, since all the thought experiments (involving light clocks) seem to assume the speed of light in that medium is $c$, not $c_m$.
 A: The equations of relativity do not hold in a medium, which selects a preferred coordinate system (the one which rests in the medium). Can you go faster than the local speed of the quasi-particles formed by the photons and the polarized atoms of the medium? Of course. If you are charged you will be generating Cherenkov radiation, too. See http://en.wikipedia.org/wiki/Cherenkov_radiation.
A: Yes, you can exceed the speed of light in a medium. In fact an everyday example of this would be a good conductor like copper. The speed of sound in copper is about 4600 m/s, but the phase velocity of electromagnetic waves in copper is a mere $v \simeq 0.4  f^{1/2}$, where $f$ is the frequency in Hz. Hence for frequencies below a GHz the speed of sound exceeds that of light in copper.
Thus information can clearly be passed faster than the speed of light in a medium, so it would be absurd to have thought experiments where information transfer was limited to that speed.
A: I'd like to look at CuriousOne's Answer in more detail, particularly his key statement:

"a medium ... selects a preferred coordinate system"

I think you may be giving too much weight and significance to the notion of light. The universal, relativistic $c$, first and foremost, is a universal constant whose existence follows from certain basic symmetry considerations, altogether independently from light, as I discuss in my answer here. Essentially $c$'s existence can be inferred by imagining what happens to Galileo's relativity when one drops the assumption of absolute time. It follows from these considerations alone that $c$ also signifies a velocity which transforms in a striking and peculiar way (i.e. is the same in all inertial frames) and historically light was experimentally the first thing found whose speed transforms in this peculiar and striking way. Of course, it was Einstein who completed the generalisation of Galileo's relativity motivated by the experimental results (the Michelson-Morley experiment) about light that were highly topical at the time. But nowadays we can understand special relativity and Lorentz invariance independently of light as simply the generalisation of Galileo's magnificent insight (the so called first relativity postulate, illustrated by the Allegory of Salviati's Ship)  by one so bold (Einstein) to imagine that time might not be absolute. It's still, nonetheless, Galileo's basic idea of symmetry that underlies everything. Your inertial frame can be truly, objectively, different from mine even though we can't see one another if we can both see the same medium passing us by.
So, given you know now that it is symmetry considerations which motivate the definition of $c$, let's look at CuriousOne's answer afresh.  You can't reason relativistically with the propagation speeds of disturbances in a medium because the medium itself throws up a "preferred frame" and thus shatters the symmetry assumptions that underlie the first relativity postulate. If we're passing through a medium, we can "watch it go by" relative to our own inertial frame and objectively measure how fast it does so. 
To complete the understanding of our error in trying to reason relativistically with the speed of light in a medium, it is worth pointing out that light in a medium is not really light but a quantum superposition of free photons and excited matter states in the medium (see my answer here for more on this description) and the delay whilst energy is fleetingly absorbed in the exited matter states accounts for why the disturbance propagates at slower than $c$ (the universal relativistic one). Free photons always move at the universal $c$ because they have a rest mass of nought. 
