It is possible for milk ducts to become blocked, especially during the first few weeks of breastfeeding until your body learns exactly what your baby needs and does not overproduce your milk supply.
If you feel that this is happening, try massaging the firm area while nursing and use a warm compress to this area while doing so. This should dislodge the plugged milk duct and allow milk to flow freely again from that area of the breast. You can also try nursing in different positions which will vary the area of the breast that is drained.
If the above instructions do not resolve the problem and you begin running a fever, having redness in any part of the breast, or feeling flu-like symptoms you should call your physician or midwife immediately.
They will look at the breast and if they think you may have an infection (mastitis) they may order antibiotics, have you continue to use heat and massage, and will usually have you continue breastfeeding to properly drain the breast. Follow all of their instructions and complete your antibiotics if they have been ordered. As always, stay well hydrated.