Can a 24 year old get breast cancer?

Yes: Very rare but possible. Many "breast lumps/tumors" in young people are benign but don't ignore an obvious palpable nodule. Get checked to be safe.

Yes: It is rare, but does occur mostly in women with a genetic predisposition to cancer. Brca mutations are typically associated with breast cancer at a young age. These do occur sporadically, though typically less then 10% of the time. Brca stands for breast cancer. We do test for this in families with a strong history of breast cancer, and we usually like to test the index case. Ask your doctor.

Rare: While possible, it would be very uncommon.