Is rudder deflection sufficient to maintain directional control on the runway at high speed, or is nose wheel steering needed?
There is some overlap in effectiveness, and therefore no real magic number, but most aircraft that don't have full time nosewheel steering will have some published airspeed at which one will be favored over the other.  For example, in the EA-6B it was 80 KIAS.  On take off roll you would hold the nosewheel steering button on the control stick down to maintain directional control with the nose wheel.  Accellerating through 80 KIAS it was a call-out to release the NWS button and rely on the rudder effectiveness to keep control up to rotation at around 145 KIAS.  The reason for this was not because the nosewheel was less effective due to less friction, (traction) but rather there was a risk of overcontrolling steering at higher speeds.