Yes, but we almost always measure c precisely using light near the visible spectrum. Rough estimates were first made based on the behavior of Jupiter and Saturn's moons (their eclipses occurred slightly too soon when the planets were near Earth and slightly too late when they were far from Earth).
Variants of a Foucault apparatus are still used and that's almost completely with visible light or near visible light.
One can also use microwaves to do clever stuff with cavity resonance. I'm not sure if there would be a noticeable energy difference.
The ideal thing would be to measure the speed of light for higher energy forms of light, like x-rays and gamma rays. But I'm not aware of any experiments that do that.
The experimental upper bound on photon mass is 10^-18 eV. The photons near visible spectrum have about 10^-3 eV, which means their relative deviation from c is of order 10^-30. Gamma would be even closer. I don't think mass of photon is measurable via speed of light.
http://www.nature.com/news/2011/110922/full/news.2011.554.html
http://arxiv.org/abs/1109.4897v1
http://usersguidetotheuniverse.com/?p=2169
http://news.ycombinator.com/item?id=3027056
Perhaps the end of the era of the light cone and beginning of the era of the neutrino cone? I'd be curious to see your probability estimates for whether this theory pans out. Or other crackpot hypotheses to explain the results.