Each landlord is different- some will be flexible with you, and some will not. Keep in mind, in most cases, the landlord has expenses to cover on the property (mortgage, insurance, upkeep), and can't afford to have it vacant, any more than you can afford double-rent.
In our current rental market, landlords do have the upper hand, as most desirable places receive multiple applications. The landlord will pick the one that is strongest financially, and the one that can move in the quickest!
The best thing to do is to time your move with very little overlap. For example, if your current lease is up July 31, try to secure a new place for move-in around July 27. That way, you only have 4 days of overlapping rent.
Nicole Fedorchek, Realtor