Yes, you can sign your lease in April to begin in August; most places start renting properties in late January for move-in the following August. This can be a major hassle for, say, a recently graduated law school applicant who won't know till April where he will be in August, and therefore stands to lose out on the best rental properties. Another bullshit aspect of the rental process in this town is nearly every major landlord has leases which run August 1st to July 27th or so. And, while you don't get your rent prorated for the partial month of July, you do end up couch-surfing for three or four days, probably with all your posessions in the back of a truck somewhere, unless your new landlord lets you claim your next apartment early and absolve them of cleaning up after the outgoing tenants.
I reiterate my small private landlord advice from before; they are less likely to be sticklers about dates and the way the rest of the town writes their leases. That, and put that aspiring legal mind to work early and make sure you read the fine print. We do have some slum lords in town too.
Edit: I should disclose, my knowledge applies strictly to campus area housing; if you don't mind living a little further out, or in Coralville (town attached to Iowa City) you can probably find more flexible lease terms.