ATLANTA (AP) — Federal health officials now say that 4,000 or more Americans likely have died from swine flu — about four times the estimate they've been using. The new, higher figure was first reported by The New York Times. It includes deaths caused by complications related to swine flu, including pneumonia and bacterial infections. Until now, the Centers for Disease Control and Prevention had conservatively put the U.S. swine flu death count at more than 1,000. Officials said this week they're working on an even more accurate calculation. The CDC says "many millions" of Americans have caught the pandemic flu virus since it first appeared in April.