Do people ever ovulate then start their period a few days later?

Usually no: That really should not happen. If it did, that would indicate a severe problem with the ovaries, essentially being close to menopause, where the ovaries cannot produce enough hormones to sustain the uterus and so the hormones drop quickly and a period comes. The bigger question is did ovulation really occur and if bleeding occurred was it really a true period or just abnormal bleeding.