Has the speed of light changed over time?
There is no meaningful way to test if the speed of light varies - that's because it's dimensionful, i.e. it's measured in units.
To see why, let's say we use units in which distance is measured in terms of multiples of the circumference of the electron's orbit in the ground state of Bohr's hydrogen atom, and the unit of time is it's orbital period. This will give you roughly 137, which is the inverse of the fine structure constant, which is defined as $e^2 \over \hbar c$. So, we can see that it isn't possible to determine whether the value of the speed of light was different, since one of the other constants in the FSC (the electron charge or the reduced Planck constant) could have changed.
However, it is meaningful to ask whether a dimensionless constant has changed, one that isn't measured in units. Some examples are the above mentioned fine structure constant, and the cosmological constant. Also, particle masses are fundamental constants - changing another constant doesn't affect them.
So, rather than asking if the speed of light varies, a better question is to ask if the fine structure constant varies (since it is dimensionless, it has no units). There have been claims that the fine structure constant may vary (here and here, among many others). However, this certainly isn't an accepted result.
For more, see the Usenet FAQ on dimensionless constants:
http://math.ucr.edu/home/baez/constants.html
Addition
Rather than varying over time, let's think of the case in which c varies over space. So, a group of scientists ventures on a rocket to a distance part of the galaxy to determine if the speed of light is different. They will need to use the same units that the earth scientists are using - we could use the above units, the vibrations of an atom for time, whatever you want. Let's say they measure a different value using the agreed units.
Now, imagine that a different group of scientists was going to test if the length of some particular rod was different in that same region of the galaxy. They decide to see how many vibrations of the cesium atom it takes light to travel the rod. Based on their experiment, they come to the conclusion that the length of the rod is larger in this other region, or that the cesium atom vibrates slightly faster.
When both groups publish their findings, they disagree - the first group tells the second group they're wrong because they based their measurements on the speed of light, which they found varies. However, group two asserts that the first group is mistaken, since they found that the length of the measuring rod and frequency of the vibrations of the cesium atom were both different.
So, you can see that asserting that a dimensionful constant has varied is meaningless - since they're ratios of other constants, it is 100 percent equally valid to say those constants varied. Not only is it impossible to determine if they have changed, but the question itself doesn't have an answer. Finding different values for dimensionful constants can be interpreted in a variety of ways. For example, you can claim that the constants in the fine structure constant had varied, not the speed of light.
It has been claimed based on astronomical observations that the unitless fine-structure constant $\alpha=e^2/\hbar c$ actually varies over time, rather than being fixed.[Webb 2001] This claim is probably wrong, since later attempts to reproduce the observations failed.[Chand 2004] Rosenband et al.[Rosenband 2008] have done laboratory measurements that rule out a linear decrease of $\alpha$ with time large enough to be consistent with Webb's results.
Webb et al. have recently made even more extraordinary claims that the fine structure constant varies over the celestial sphere.[Webb 2010] Extraordinary claims require extraordinary proof, and Webb et al. have not supplied that; their results are at the margins of statistical significance compared to their random and systematic errors.
Even if their claims are correct, this is not evidence that $c$ is changing, as is sometimes stated in the popular press. If an experiment is to test whether a fundamental constant is really constant, the constant must be unitless.[Duff 2002] If the fine-structure constant does vary, there is no empirical way to assign blame to $c$ as opposed to $\hbar$ or $e$. John Baez has a nice web page discussing the unitless constants of nature.
J.K. Webb et al., 2000, "Further Evidence for Cosmological Evolution of the Fine Structure Constant," http://arxiv.org/abs/astro-ph/0012539v3
J.K. Webb et al., 2010, "Evidence for spatial variation of the fine structure constant," http://arxiv.org/abs/1008.3907 ; Phys. Rev. Lett. 107, 191101 (2011)
H. Chand et al., 2004, Astron. Astrophys. 417: 853, http://arxiv.org/abs/astro-ph/0401094 ; See also http://arxiv.org/abs/0711.1742 , http://arxiv.org/abs/0905.1516
Srianand et al., 2004, Phys.Rev.Lett.92:121302, http://arxiv.org/abs/astro-ph/0402177
Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093
Baez, http://math.ucr.edu/home/baez/constants.html
Rosenband et al., 2008, 319 (5871): 1808-1812, http://www.sciencemag.org/content/319/5871/1808.abstract