That would be the 60D's flash sync speed. When you have a flash connected you can not get a shorter shutter speed than that without enabling high speed sync. That is because of the way the shutter works - even at a 1/8000 shutter speed, the shutter "rolls" across the image, taking about 1/250 second. Each individual part is only exposed for your 1/8000 but it takes that time to roll across the image. When you use flash, the exposure has to have the whole frame being exposed at the same instant, and that's only possible at a speed longer than what it takes the shutter to roll the image.
If you don't have a flash, the camera surely thinks you do, and something else is wrong.