Sliding friction must, logically always be less than static friction. This becomes obvious when you look at it in terms of forces rather than coefficients.
Consider a block resting on a flat surface. You apply a small force to one side of the block to try to slide it along. Initially it doesn't move as the reaction force provided by friction increases in line with the force you exert.
Say that applying 10N of force is just enough to overcome static friction and get the block moving. Now if the sliding friction becomes greater the 10N force is no longer enough to overcome it and so it stops and your back to a static condition.
Another way to look at this is that friction is a reaction force so it can never be greater than the force applied otherwise you would be effectively getting work from nowhere.
However it is certainly possible to have materials whose friction properties change as a result of friction, for example carbon-carbon brakes are most effective at quite high temperatures.
There is also viscous drag which is proportional some function of speed although it is not really very meaningful to talk about static friction in this context.