When will my baby be able to handle large pieces of food?

Depends. Most infants can start to tolerate more solid foods (such as bits of pasta) around 9 months of age. This often occurs around the same age the baby develops a good pincer grasp. However, some babies have problems which delay the timing (eg. Reflux, feeding aversion or gagging). Also many premature babies need extra time to develop this skills. It is best to consult your doctor on the right timing.
In a few years. Babies and toddlers choke on large pieces of food. They can even choke on small pieces of food. This is why baby foods are mashed up or pureed. Toddler foods are soft so that they break up easily when a toddler presses the food with his tongue or gums.
Probably Not. Babies and toddlers up to the age of 2-3years old are at much highr risk of choking, so that care must be taken with large chunks of food.Think of a cheerio, which if put in the baby's mouth and not chewed is small enough to not block the windpipe and will dissolve. This should serve as the model for feeding table foods to the baby.
As late as... ...Possible. Fatal choking can occur at any age, but is most common below 3.