"Mr. and Mrs. Borden took sick on Tuesday evening, following a supper of warmed-over swordfish, which had been fried fresh that day for the noontime meal. Lizzie didn't become ill until Wednesday morning, at which time the family had pork steak and johnnycakes for breakfast." (p.325+). We know that Abby went to see Dr. Bowen that same morning, complaining that the baker's bread and/or cake could have been poisoned. He gave her castor oil to be taken with a little wine. Later that morning, he went to check on Abby and ran into Andrew, who admitted, "he felt a little heavy, and did not feel just right, but said he did not think he needed any medicine." By Thursday morning, they were already sick, most likely from the swordfish or something they'd eaten earlier.
Of course, reading about how the stores and/or butchers stored or displayed their meats, it's no wonder the whole town didn't get sick!