Yes if your alternator is bad and was not charging your battery up while driving the car can still start and run if you have someone jump your cars battery or if you charge the battery up.
The alternator won't prevent the car from starting but if it doesn't charge the battery then the battery will be drained while driving and cause the car to not start.
You can easily test your alternator to see if it's charging by starting your car up and let it idle and then disconnect the positive battery cable and if the car dies then the alternator is bad.
If the car continues to run then the alternator is charging and producing voltage to keep the engine running.