Does the TI CC2540 support Bluetooth High Speed protocol?
No it doesn't.  Bluetooth 3.0 or Bluetooth 4.0 does not imply High Speed support. Bluetooth 4.0 doesn't necessarily even imply backwards compatibility with Bluetooth 3.0 devices. iPhone 4S has hardware that supports High Speed, but the software only supports Bluetooth 4.0, not high speed.

For a lot more info, see http://chipscoop.blogspot.com/2012/08/what-is-bluetooth-smart.html