No. The state court could not bring charges again as that would constitute double jeopardy.
In theory, if a federal offense is involved the US Attorney could bring charges arising out of or relating to the same incident. This would not constitute double jeopardy as two different "soverigns" would be involved (the state as the first "soverign" and the federal government as the second). Double jeopardy contemplates charges being brought by the same governmental authority for the same offense.
From the very little I know about the Anthony matter (through the media), I have no reason to suspect that the Federal authorities will get involved.