Yes, you can usually start a car with a bad alternator. The alternator's job is to *recharge* the battery while the engine is running. The battery itself provides the power to start the engine. So, if your battery has enough charge, a faulty alternator won't prevent the car from starting.
However, once the engine is running with a bad alternator, the battery won't be recharged, and eventually, the battery will drain completely, leaving you unable to start the car again.
