In Quebec, there is a maximum insured by the state and it start 180 days after the event. So if you hand up breaking your ribs/hands and stop working for 4 months, you never get this. My private insurances cover from 60 days up to the 180 days at 80% and then goes down to 60%. The effect of the state insurances is to lower the cost of the private insurances.
Yes this is what happens in Italy. You pay almost 30% of what you make in a compulsory fund for retirement and disability pension and when you need it, the state pays you.