If my baby's neck is slightly swollen, does that mean he'll get sick soon?

Possibly. The main cause of a swollen neck is enlarged lymph nodes. Lymph nodes can be enlarged for several reasons. Lymph nodes are one of the bodies defenses and enlarge in response to exposure to allergens, viral and bacterial infections. Rarely they can be enlarged because of a malignant process.
Possibly. If you notice a swelling on one side of your baby's neck, it could be a swollen lymph node. If the baby is irritable, has fever, is showing other signs of illness, or if you are concerned, have this checked by your pediatrician. Sometimes lymph nodes swell in response to infection or irritation.