The speed of light is constant for all observers relative to the observer. No one will percieve light as travelling slower or faster than approx. 3.0x10^8 m/s. Thus it would be possible for us to be travelling near the speed of light without being aware of it. But relative to what? Is there some "stationary" reference that we can relate our movement to?
Although, I don't believe that we've found any stellar body to be moving away from us (or us moving away from it) at that speed, which is measured by the doppler effect: z = approx(v/c). if we were moving near the speed of light, say .99999c, the z value would be .99999. No z value has ever come that close to 1. So I'd say it's unlikely that we are moving that fast, at least not to anything we can see in the sky.
