A plugged catalytic converter will only in extreme cases prevent you from driving your car. You can usually drive your car indefinitely with a partially plugged converter, though you will see a drop in performance. The converter might become completely fused almost immediately or over time and then would require immediate replacement.
It Can Burn Through Over Time
When your engine is burning oil or running too rich, your catalytic converter's inner fibre may be "burned through." If the converter is burnt through you can still drive the car as long as you want, but will notice some smoke and a smell.
Most catalytic converter problems do not prevent your car from functioning, but if a catalytic converter is completely plugged you should not drive your car at all. If you live in an area with smog checks, you will not be able to legally drive your car without having the damaged converter replaced.