This is called a dispensational theology. Dispensationalists hold that though out history, God has related to man in different ways, and that miracles were in a past "dispensation".
I am not a dispensationalist. I personally have met at least five people who have been raised from the dead, and I was there when one of them was (See my hub, "You will do Greater Things than I".)
I also know people who have been healed of documented medical diseases and conditions. Same thing with demons.
So, yes I believe we are called to do it today. And if it is impossible, ... that means we have to have God directly involved.