At some 0.999999999c (9 nines) white light (540 THz, center of our visible spectrum) becomes gamma (above 10^19 Hz). CMB is less of an issue, as it's both weaker and of lower frequency than starlight (it would be five orders of magnitude below gamma at this speed).
Interstellar matter, however, would get nasty much sooner than that. It doesn't really matter how dense the void is, as even 1 proton per cubic km would be enough to ensure several collisions per second (a 10-meter cross-section ship going 0.999999999c plows through 23 km³ every second, so it's bound to get hit a lot).
A single proton impacting a ship's erosion shield at 0.999999999c releases an energy of roughly 75 mJ - not that much, but considering we're talking about a single proton... that much energy concentrated in a small area like that can easily knock a couple dozen atoms off the erosion shield. A day or two at this rate, and the bow starts to get all ugly.
Note that real interstellar medium is much, much more dense than this - several particles per cubic meter (several billion per cubic km). At these speeds it might even work like aerobraking