How Soon do I Need to Sign an Apartment Lease?

I’m going to grad school here in the fall, so I’ve already started looking at apartments around the school (since I won’t have a car, it’s imperative that I live within walking distance), and I’ve noticed that none of them have very many rooms left. Will more open up towards graduation, since they’ll be finding out which students won’t be renewing their leases? Can I wait that long to choose an apartment, or should I jump on one right now? I’m also going to visit campus at the end of the month, so it’s possible that I could visit apartments then, as well…but can I wait that long?