If a piece of tooth falls off of a tooth with a filling after eating something does that mean something was wrong with the tooth?

Fillings can reduce the solidity/structure of a tooth and chewing can bring forces to the tooth that will cause a crack/fracture Once the crack occurs a piece can break off. The bigger the filling & loss of tooth structure the more likely it is to crack/break. Fillings can delay the need for a crown forever or just a few years. I would not assume your natural teeth are of poor structural integrity

Yes, it means the tooth structure is somehow compromised either by decay or a fracture. See your local dentist.

It means the remaining tooth material is compromised. Fractures can be caused by the enamel being undermined by decay, large restorations, trauma, eating hard foods like ice, hard candy, etc), parafunctional habits, etc. Please see your Dentist as soon as possible.