No the Fed can't go BK....however the mechanism for their liquidly CAN vanish. The bids at Treasury Auctions can and are drying up.
It can sort of go bankrupt if it debases the currnency to the point where creating anymore is just meaningless and thus has no effect at the margin. But don't worry will take another few years or so. With Roman coins or Weimar paper Deutshmarks the cost of producing probably beacme greater than the money the represented.