Solar systems are capable of working on overcast and cloudy days, just not at full capacity. In low light conditions such as early in the morning or when it is overcast, solar panels will still convert light into power for your home, just at a reduced rate.
For example, a 4kW system may only be outputting 2kWs of power on a cloudy day. There is a point where the energy produced is no longer strong enough to power the inverter and the system will turn off. This happens every evening and perhaps during an especially dark storm