Probably not

A group velocity faster than $c$ does not mean that photons or neutrinos are moving faster thsn the speed of light.

This is the conclusion of Fast light, fast neutrinos? by Kevin Cahill⁽¹²⁾. He start his briefly analisys from some experimental observations of superluminal group velocity. In these experiments researchers measure a speed of light faster and slower than $c$ in vacuum. The first observation was occured in 1982⁽¹⁾, but an interesting collection of work in this subject is in Bigelow⁽⁷⁾ and Gehring⁽¹¹⁾. Experimentally when some pulses journey into a highly dispersive media occur some exotic effects. One of these is the observation of a negative group velocity, that coincides with a superluminal speed.
In Bigelow's and Gehring's works wasn't a really theoretical explenation. For example Bigelow proposed the following explaination:

(...) as the combination of different absorption cross sections and lifetimes for Cr³⁺ ions at either mirror or inversion sites within the BeAl₂O₄ crystal lattice. The superluminal wave propagation is produced by a narrow “antihole” [612 Hz half width at half maximum (HWHM)] in the absorption spectrum of Cr³⁺ ions at the mirror sites of the alexandrite crystal lattice, and the slow light originates from an even narrower hole (8.4 Hz) in the absorption spectrum of Cr³⁺ ions at the inversion sites.

They also considered

(...) the influence of ions both at the inversion sites and at the mirror sites. In addition, the absorption cross sections are assumed to be different at different wavelengths.

The arrows indicate the locations of ion sites that have mirror or inversion symmetry. On the right, the corresponding energy-level diagrams for Cr³⁺ ions at the different sites are shown.

Gehring and collegues, instead, write that:

(...) theory predicts that the peak of the transmitted pulse will exit the material before the peak of the incident pulse enters thematerial, and furthermore that the pulsewill appear to propagate in the backward directionwithin the medium^{(2, 3, 4)}.

In (a) experimentally measured group delay vs. crystal rotation;
in (b) the weak value $\left [A_\beta \right ]_W$ vs frequency⁽⁹⁾

Another theoretical explenation was proposed in Brunner⁽¹⁰⁾ where researchers proposed the quantum formalism of weak values developed by Aharonov and Vaidman. The connection between weak values and superluminal velocity was examined and proofed by Daniel Solli^{(8, 9)}, who proposed the following experiment:

Our experimental system consists of a slab of highly birefringent two-dimensional (2D) photonic crystal and a linear polarizer, placed in series. The photonic crystal has fundamental and second-order photonic band gaps in the regions of 10 and 20 GHz, and displays strong birefringence with very high transmission in the frequency range between the two gaps⁽⁶⁾. The crystal itself is an 18-layer hexagonal array of hollow acrylic rods (outer diameter 1/2”) with an air-filling fraction (AFF) of 0.60. The crystal was constructed using a method which we have previously described⁽⁵⁾

In particular the group velocity is defined by \[v_g = \frac{L}{t_f + < t > }\] where $L$ is the lenght of the medium crossed by the pulse, $t_f$ the free propagation time and $< t >$ the mean time of arrival once the free propagation has been subtracted⁽¹⁰⁾, that it can be expressed like a function of the weak value $W$: \[< t > = \frac{\delta \tau}{2}{\text Re} W\] with $\delta t$ the temporal shift between the two eigenmodes⁽¹⁰⁾.
Using the weak formalism Antonio Mecozzi and Marco Bellini⁽¹³⁾ try to explain the recently OPERA's superluminal neutrinos imagining a similar effect also for these particles. Their theoretical calculation was reproduced by Berry, Brunner (the same of the experimental paper in note 10), Popescu and Shukla⁽¹⁴⁾, but they also confront theoretical result with experimental observations. And finally they find that the effect of the weak formalism is too... weak in order to explain OPERA's observation.

---

In conclusion, after a brief story about superluminal signals in optics, sir Michael Berry (I hope to write something in future about his famous geometrical phase). with Brunner et al., quest: Can superluinal neutrino speeds be explained as a quantum weak measurement?
An the answer is in the abstract:

Probably not.

And it is defined the best abstract ever by Improbable research. Indeed, I remember that Berry winned IgNoble in 2000 with Andre Geim.

---

About neutrinos: there's a lot of news (i hope to write something about the preprints proposed). For example paper by Ronald van Elburg, in which author supposed that OPERA collaboration forgotten to examine some lorentzian effects.
And you also just read the preprint by Glashow and Cohen in which the two theorists refute the superluminal interpretation of OPERA results. Well, some days ago ICARUS, an experiment designed and directed by Carlo Rubbia based on Gran Sasso National Laboratory (the same laboratory of OPERA) confirmed in a preprint Glashow's and Cohen's ideas.
Finally, Torino's INFN published a page with all superluminal preprints. And you can see also my little storify with some preprints about the question.

---

(1) S. Chu, S. Wong (1982). Linear Pulse Propagation in an Absorbing Medium Phys. Rev. Lett. 48, 738. DOI 10.1103/PhysRevLett.48.738
(2) R. Y. Chiao (1993). Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations Phys. Rev. A 48, R34. DOI 10.1103/PhysRevA.48.R34
(3) M. Ware, S. A. Glasgow, and J. Peatross (2001). The Role of Group Velocity in Tracking Field Energy in Linear Dielectrics, Opt. Express 9, 506-518. DOI 10.1364/OE.9.000506 (pdf)
(4) M. Ware, S. A. Glasgow, and J. Peatross (2991). Energy Transport in Linear Dielectrics, Opt. Express 9, 519-532. DOI 10.1364/OE.9.000519
(5) J. M. Hickmann, D. Solli, C. F. McCormick, R. Plambeck, R. Y. Chiao (2002). Microwave measurements of the photonic band gap in a two-dimensional photonic crystal slab, J. Appl. Phys. 92, 6918. DOI 10.1063/1.1518162
(6) D. R. Solli, C. F. McCormick, R. Y. Chiao, J. M. Hickmann (2003). Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates, Optics Express 11, 125. DOI 10.1364/OE.11.000125
(7) Bigelow, M. (2003). Superluminal and Slow Light Propagation in a Room-Temperature Solid Science, 301 (5630), 200-202 DOI: 10.1126/science.1084429
(8) Solli, D., McCormick, C., Ropers, C., Morehead, J., Chiao, R., & Hickmann, J. (2003). Demonstration of Superluminal Effects in an Absorptionless, Nonreflective System Physical Review Letters, 91 (14) DOI: 10.1103/PhysRevLett.91.143906 (arXiv)
(9) Solli, D., McCormick, C., Chiao, R., Popescu, S., & Hickmann, J. (2004). Fast Light, Slow Light, and Phase Singularities: A Connection to Generalized Weak Values Physical Review Letters, 92 (4) DOI: 10.1103/PhysRevLett.92.043601 (arXiv)
(10) Brunner, N., Scarani, V., Wegmüller, M., Legré, M., & Gisin, N. (2004). Direct Measurement of Superluminal Group Velocity and Signal Velocity in an Optical Fiber Physical Review Letters, 93 (20) DOI: 10.1103/PhysRevLett.93.203902 (arXiv)
(11) Gehring, G. (2006). Observation of Backward Pulse Propagation Through a Medium with a Negative Group Velocity Science, 312 (5775), 895-897 DOI: 10.1126/science.1124524
(12) Kevin Cahill (2011). Fast Light, Fast Neutrinos? arXiv: 1109.5357
(13) Antonio Mecozzi, Marco Bellini (2011). Superluminal group velocity of neutrinos arXiv: 1110.1253
(14) M. V. Berry, N. Brunner, S. Popescu, & P. Shukla (2011). Can apparent superluminal neutrino speeds be explained as a quantum weak measurement? arXiv: 1110.2832v1