It depends on the fuel and the condition of the catalytic converter. If the fuel is too high in sulfur, it can cause the smell. If the cat is bad, it can cause the smell. Like others have said try changing your brand of fuel. Even if it's not a no-name brand, change to something else. Also try some fuel injector cleaner.
Sulphur will render a catalytic converter completely useless pretty quickly. High sulphur fuels will, over time, gradually degrade its effectiveness and the rotten egg exhaust smell is typically the sign that the cat is toast. If it's been getting worse over the past few tanks of gas, the cat is probably a goner or close to it. If it just started with a recent fill-up, get that stuff out of your tank. Injector cleaner won't help with this kind of problem. One tank of sulphur-laden fuel, which is highly unlikely but possible, can destroy the cat. Once the sulphur has infiltrated the catalyst media, it is permanently altered and will not be effective again no matter what kind of cleaning you do. It will have to be replaced.
On an OBD-1 vehicle, there is probably no sensor after the cat to determine its efficiency, so you probably won't see any console messages about exhaust system problems. If you got it emissions tested, though, and the cat is bad, it will fail miserably. On an OBD-II vehicle, 96 or newer basically, there should be a sensor after the cat and it would set a DTC indicating reduced catalytic converter performance. It would show up as a SES light, a console message, or both.