You can get addicted when you use too many drugs. As an addict, you suffer withdrawal symptoms only when you run out of supplies. Many substances in the Fallout games are addictive. Each chem has a chance of addiction. One of the most addictive drugs is jet.
For Fallout 1 & 2, withdrawal effects occur after becoming addicted and going for some amount of time without using that chem. For non-jet addictions, addiction lasts 7 days from the onset of the withdrawal effects (nine days of non-use), while jet addiction lasts forever without the Jet Antidote.