Yes you can get sick eating old broccoli that has gotten bacteria or mold on it.
The old broccoli can lead to food poisoning and cause you to have symptoms such as cramping, bloating, gas, fever, diarrhea or vomiting.
When broccoli is going bad the bright green color will begin to turn yellow and it's crisp texture will become limp.
If the broccoli is going bad it's best to throw it out and not risk getting sick over it.