Can your gums around a tooth become swollen if you eat something sharp, like chips, if it pokes you there?

Yes, but... There are other things that could have caused the gum swelling so it's best to have it checked out by your dentist.
Definitely. Chips can scratch, poke and even lodge between the gums and the teeth. Make sure you brush and floss the area if you suspect that the chip has lodged. If in doubt, get to the dentist.
Yes. Tnis can cause some pain. If you get a piece of food stuck between your teeth or under your gums they can become sore, red and swollen.. Try flossing the get it out. If this does not help,then talk to your parents about seeing a dentist.