Subway recently announced that within the next decade they will move toward serving only protein from animals never treated with antibiotics in their more than 25,000 locations nationwide. This isn't only those antibiotics used in human medicine, this also includes those that are only used when an animal is ill. Many times, an illness can be cured with an antibiotic, much the same way that going to the doctor and getting something can help you or your child. Antibiotics help keep animals healthy because, just like people, they do catch a cold or infection from time to time.
There are a lot of procedures in place on the farm to keep antibiotic residues out of the food supply. On most (if not all--I haven't seen every antibiotic on the market) labels on pharmaceutical containers, there are meat and milk withdrawl times. This represents the amount of time the residue takes to exit the body. Let's pretend there's a 48 hour milk withdrawl and 10 day meat withdrawl on a particular pharmaceutical. Cows (and animals in general) metabolize things differently, so while a milk withdrawl time may be 48 hours, dairy farmers generally test the milk before putting it back into the bulk tank (refrigerated tank on the farm where the milk stays chilled). This withdrawl time starts at the time of the last treatment. One cow may be clean 36 hours later and another 52 hours later, it's not always the same, which is why testing the milk is important. Meat withdrawl times are longer than those for milk. The 10 days begins at the last treatment, just like for milk, and says the animal should not be slaughtered until that period is up. It's not a bad idea to add some time to that if the cow is going to be culled just to be safe.
I have a lot of confidence in our food supply. After being heavily involved in the dairy industry for most of my life, both on the farm and within industry organizations and companies, I've seen firsthand the practices farmers and those that work with them use to keep the food supply safe. Repercussions are stiff when something goes wrong, so farmers are very careful. Ship a load of milk with antibiotic residue and not only will that load be returned to dump in the manure lagoon, but you will also end up buying the entire load. Yes, that includes all of the milk on the truck, not just yours. At around $10,000 (varies depending on the milk price) a load, that's something to take seriously.
There are people like veterinarians, nutritionists and numerous others that provide guidance to farmers to help them provide the best care possible for their animals and also to keep the food supply safe.
I don't know as much about the meat side of things when it comes to slaughter plants, but I do know that USDA generally has inspectors testing a percentage of the meat that comes through the doors. Creating a safe food supply starts at the farm level.
I'm thankful I live in a country where I don't have to worry about whether the meat or milk I buy in the grocery store or at a restaurant is safe or whether it may have some antibiotic residue in it.