Would a faulty alternator prevent the car from starting?

Should I be able to isolate the alternator cables and then start the car and keep it running on the battery alone?

Reason. Changed out radiator. Alternator has wonky bearing. While I had all that front end stuff off, I pulled alternator off to take it to repair shop to get quote for refurbishment. Due to some confusion over the specifics of my alternator I just put it back on the car as it was. Car will now not start - turns over wants to start - a few coughs etc but no go.