Special relativity says that as you accelerate, say, a spaceship, towards the speed of light its inertial mass increases so it gets harder to push it any faster. At the speed of light its inertial mass is infinite so you can't increase its speed at all. Hence relativity predicts a speed of light speed limit. However, MiHsC makes a slight correction to this. The wavelength of the Unruh radiation that causes inertia in MiHsC lengthens as the acceleration reduces which means that, for the spaceship case above, as the speed levels off and acceleration tends to zero near the speed of light, the Unruh waves making up its inertia exceed the Hubble-scale and cannot be observed. This means, using the philosophy of Mach that special relativity itself was based on, these waves should dissapear, and the spaceship's inertial mass should reduce. Indeed, putting MiHsC and relativity together (in a very preliminary way) you can show that there remains a residual relativity-proof acceleration of 2*(speed of light)^2/(Hubble scale) even at the speed of light: this is the minimum acceleration allowed by MiHsC. Interestingly this is close to the cosmic acceleration that has recently been observed and is usually explained in an ad hoc manner by dark energy.
For such a claim of course, far more direct evidence needs to be found. There are ways in which observations of quantum systems demand non-locality and superluminal information transfer (Bell's inequalities), but my favourite possibility at the moment involves the more direct evidence of galactic jets. Looking at the movement of blobs of light within the jets streaming out along the spin axes of galactic cores and quasars, and knowing the distance of these objects, it is possible to show that these blobs appear to move faster than light (eg: Porcas, 1983, Biretta, 1999). Before we get too excited, Martin Rees (1966) showed that light-emitting objects moving at sublight speeds can appear to travel faster than light if they are moving at a small angle to our line of sight. However, that being so, one would expect the jets that show faster than light speeds to all be apparently 'shorter' since they should be pointing towards us, but it has been shown that they are not shorter on average than all the other ones, which implies that they are not on average close to our line of sight. A particular case is M87 (Biretta et al., 1999). The blobs of light in its jet are moving at six times the speed of light. To explain this away as the Rees effect one would need this jet to be within 20 degrees of our line of sight, but an analysis suggests that this angle is 44-64 degrees, and to get it within 20 degrees would 'present several problems' (Biretta et al., 1999).
I know this is a horrifically complex area to get into, and causality will have to be thought about too which means that thinking about it is rather like taking an axe to the floor one is standing on, but I do think this is important, doubly so since I'm one of the few arguing that FTL (Faster Than Light) is possible. I've had some problems publishing anything on this. I've submitted papers, and I gave a talk on MiHsC and FTL at the 100 Year Starship Symposium in Orlando in 2011, and my talk was filmed and was supposed to be made available. Nothing happened, and nothing happened to the paper I sent to them either, so I'm very glad to finally have a chance to publish something on FTL in my book.
Biretta, J.A., et al., 1999. Hubble space telescope observations of superluminal motion in the M87 jet. The Astrophysical Journal, 520, 2, 621-626. http://adsabs.harvard.edu/abs/1999ApJ...520..621B
Porcas, R., 1983. Superluminal motion - astronomers still puzzled. Nature, 302, 753-754. http://adsabs.harvard.edu/abs/1983Natur.302..753P
Rees, M.J., 1966. Appearance of relativistically expanding radio sources. Nature, 211, 468-470.
McCulloch, M.E., 2014. Physics from the Edge: a new cosmological model for inertia. World Scientific Publishing.