If a swollen lymph node is painful, does that mean it isn't cancer?

No: Not necessarily. A lymph node that rapidly enlarges can be painful.

Not necessarily: Most swollen lymph nodes are not cancerous, they are reactive or inflammatory. By that fact most lymph nodes that are tender will be benign. Unfortunately, a small percentage will still be malignant. If you are worried please see your doctor.

Monitoring Centers: Lymphatic system absorbs/carries things too big to go into the arteries/veins like bacteria eaten by white cells and digested food. Lymph nodes "taste" the lymph for anything bad like bacteria/viruses/cancers, etc. When it detects a problem, it alerts the immune system and your body reacts. They enlarge in response to the inflammation they create when they react. They shrink when all is well.