I am from Australia, so my answer is based on my experience over here, however it should be similar for the USA.
Generally, what determines both the price of houses/apartments and the rents for them is supply and demand.
When there is high demand and low supply prices (or rents) generally go up. When there is low demand and high supply prices (or rents) usually go down.
What can sometimes happen when house prices go down, is that the demand can drop but so can supply. As the prices drop, developers will make less money on building new houses, so stop building new houses. Other developers can go bankrupt.
As less people (including investors) are buying houses, and more people (including investors) try to sell their existing houses, there will be more people looking to rent and less rental properties available to rent. This produces a perfect storm of high demand and low supply of rental properties, causing rents to rise strongly.
When the property prices start to go up again as demand increases, there is a shortfall of new properties being built (due to the developers not building during the downturn). At this time developers start to build again but there is a lag time before the new houses can be completed. This lack of supply puts more pressure on both house prices and rents to go up further.
Until equilibrium between supply and demand is realised or an oversupply of rental properties exists in the market, rents will continue to rise.