Can you have sore enlarged breasts and not be pregnant or gaining weight?

Yes. Hormonal changes cause an increase in the glandular portions of the breast with fluid retention.
Yes, manyncauses. Estrogen levels peak mid cycle causing breast duct enlargement whereas Progesterone peaks just prior to menstruation and is associated with lobular growth. This will vary from one woman to another and can be related to family history/caffeine/weight. If atypical for you consider pregnancy test. Birth control may help to ease. Exogenous hormones or drugs may also be responsible.